PHYSICAL  SIGNIFICANCE 
OF  ENTROPY 

OR  OF  THE  SECOND  LAW 


BY 

J.  F.  KLEIN 

Professor  of  Mechanical  Engineering, 
Lehigh  University 


NEW    YORK 

D.  VAN  NOSTRAND  COMPANY 

23   MURRAY  AND    27   WARREN   STREETS 

1910 


COPTRIGHT,   1910, 
BY 

JOSEPH   FREDERICK   KLEIN 


THE  SCIENTIFIC 
•OBERT   DRUMMOMO  AND  COMPANY 
BROOKLYN,   N.  Y. 


PREFACE 


IN  this  little  book  the  author  has  in  the  main  sought  to  present 
the  interpretation  reached  by  BOLTZMANN  and  by  PLANCK.  The 
writer  has  drawn  most  heavily  upon  PLANCK,  for  he  is  at  once 
the  clearest  expositor  of  BOLTZMANN  and  an  original  and  important 
contributor.  Now  these  two  investigators  reach  the  result  that 
entropy  of  any  physical  state  is  the  logarithm  of  the  probability 
of  the  state,  and  this  probability  is  identical  with  the  number 
of  "complexions"  of  the  state.  This  number  is  the  measure  of 
the  permutability  of  certain  elements  of  the  state  and  in  this  sense 
entropy  is  the  "  measure  of  the  disorder  of  the  motions  of  a  system 
of  mass  points."  To  realize  more  fully  the  ultimate  nature  of 
entropy,  the  writer  has,  in  the  light  of  these  definitions,  interpreted 
some  well-known  and  much-discussed  thermodynamic  occurrences 
and  statements.  A  brief  outline  of  the  general  procedure  followed 
will  be  found  on  p.  3,  while  a  fuller  synopsis  is  of  course  given 
in  the  accompanying  table  of  contents. 

J.  F.  KLEIN. 
LEHIGH  UNIVERSITY,  October,  1910. 

iii 


257815 


TABLE  OF   CONTENTS 


INTRODUCTION 

PAGE 

Purpose,  acknowledgments,  the  two  methods  of  approach  and  outline  of 

treatment. .  i 


PART  I 

THE    DEFINITIONS,     GENERAL    PRELIMINARIES,    DEVELOPMENT,     CURRENT 
AND  PRECISE   STATEMENTS   OF   THE  MATTERS   CONSIDERED 


SECTION  A 
(i)  The  "  state  "  of  a  body  and  its  "  change  of  state  " 5 

The  two  points  of  view;  the  microscopic  and  the  macroscopic  observer; 

the  micro-state  and  macro-state  or  aggregate 5 

The  selected  and  the  rejected  micro-states;  the  use  of  the  hypothesis  of 

"  elementary  chaos  " 7 

PLANCK'S  fuller  description  of  what  constitutes  the  state  of  a  physical 

system 10 

(2)  Further  elucidation  of  the  essential  prerequisite,  "  elementary  chaos" 

Sundry  aspects  of  haphazard n 

BOLTZMANN'S  service  to  science  in  this  field  and  his  view  of  what  con- 
stitute the  necessary  features  of  haphazard 12 

BURBURY'S  simplification  of  haphazard  necessary  and  his  example  of 

"  elementary  chaos  " 15 

Haphazard  as  expressed  by  a  system  possessing  an  extraordinary  number 
of  degrees  of  freedom 17 


CONTENTS 


(3)  Settled  and  unsettled  states;  distinction  between  final  stage  of  "  elemen- 
tary chaos  "  and  its  preceding  stages 18 

Each  stage  has  sufficient  haphazard;  examples  and  characteristics  of  the 
settled  and  unsettled  stages  of  "elementary  chaos";  all  micro- 
states  not  equally  likely;  the  assumed  state  of  "  chaos  "  does  not 
eliminate  adequate  haphazard;  two  anticipatory  remarks 19 

SECTION  B 

CONCERNING    THE    APPLICATION     OF    THE    CALCULUS    OF    PROBABILITIES 

(i)  The  probability  concept,  its  usefulness  in  the  past,  its  present 

necessity,  and  its  universality 22 

Popular  objection  to  its  use;  BOLTZMANN'S  justification  of  this  concept; 
its  usefulness  in  the  past  and  in  other  fields;  some  of  its  good  points; 
the  haphazard  features  necessary  for  its  use 23 

(2)  What  is  meant  by  probability  of  a  state  ?    Example 27 


SECTION  C 

(i)  The  existence,    definition,    measure,  properties,  relations  and  scope 

of  it 'reversibility  and  reversibility 29 

Inference  from  experience;  inference  from  the  H-theorem  or  calculus  of 
probablities;  definitions  of  irreversible  and  reversible  processes; 
examples  of  each 30 

(2)  Character  of  process  decided  by  limiting  states 32 

Nature's  preference  for  a  state;  measure  of  this  preference 33 

Entropy  both  the  criterion  and  the  measure  of  irreversibility 33 

(3)  All  the  irreversible  processes  stand  or  fall  together 34 

(4)  Convenience  of  the  fiction,  the  reversible  processes 35 

Entropy  the  only  universal  measure  of  irreversibility.     Outcome  of  the 

whole  study  of  irreversibility 36 


CONTENTS  rii 

SECTION  D 

PAGB 

(1)  The  gradual  development  of  the  idea  that  entropy  depends  on  proba- 

bility or  number  of  complexions 37 

Why  it  is  difficult  to  conceive  of  entropy.  Origin  and  first  definition 
due  to  CLAUSIUS;  some  formulas  for  it  available  from  the  start.  Its 
statistical  character  early  appreciated;  lack  of  precise  physical  mean- 
ing; its  dependence  on  probability;  number  of  complexions  a  synonym 
for  probablity 37 

(2)  PLANCK'S  formula  for  the  relation  between  entropy  and  the  number 

of  complexions 40 

Certain  features  of  entropy 41 

SECTION  E 

Equivalents  of  change  of  entropy  in  more  or  less  general  physical  terms 

or  aspects 41 

Not  surprising  that  its  many  forms  should  have  been  a  reproach  to  the 

second  law 41 

General  principles  for  comparing  these  aspects.  Various  aspects  of 
growth  of  entropy  from  the  experiential  and  from  the  atomic  point 
of  view , 1 42 

SECTION  F 

More  precise  and  specific  statements  of  the  second  law 44 

General  arrangement  and  the  principles  for  comparison 44 

Ten  different  statements  of  the  law  and  comments  thereon 44 

PART  II 

ANALYTICAL  EXPRESSIONS  FOR  A  FEW  PRIMARY  RELATIONS 

Procedure  followed 48 

SECTION  A 

MaxwelVs  law  of  distribution  of  molecular  velocities 48 

Outline  of  proof,  illustration,  and  consequences  of  this  law 48 


Tiii  CONTENTS 

SECTION  B 

PAGE 

Simple  analytical  expression  for  dependence  of  entropy  on  probability    53 

PLANCK'S  derivation;  illustration,  limitations,  consequences,  features  and 

comments 53 

SECTION  C 

Determination  of  a  precise,  numerical  expression  for  the  entropy  of 

any  physical  configuration 56 

BOLTZMANN'S  pioneer  work,  PLANCK'S  exposition,  and  the  six  main  steps.     56 

Step  a 

[  probability  or  number  ] 
Determination  of  the  general  expression  for  the  \ 

[        of  complexions        J 

of  a  given  configuration  of  a  known  aggregate  state 57 

Step  b 

Determination  of  the  general  expression  for  the  entropy  S  of  a  given  con- 
figuration of  a  known  aggregate  state *     63 

Step  c 

Special  case  of  (6),  namely,  expression  for  the  entropy  S  of  the  state  of 

thermal  equilibrium  of  a  monatomic  gas 63 

Step  d 

Confirmation,  by  equating  this  value  of  S  with  that  found  thermodynamic- 

ally  and  then  deriving  known  results 64 

Step  e 

PLANCK'S  conversion  of  the  expressions  of  (b)  and  (c)  into  more  precise 

ones  by  finding  numerical  value  of  k 66 

Stepf 

Determination  of  the  dimensions  of  the  universal  constant  k  and  there- 
fore also  of  entropy  in  general 67 


CONTENTS  ix 

PART  III 

THE   PHYSICAL    INTERPRETATIONS 

SECTION  A 
Of  the  simple  reversible  operations  in  thermodynamics 

PAGE 

Isometric,  isobaric,  isothermal,  and  isentropic  change -69 

SECTION  B 
Of  the  fundamentally  irreversible  processes 

Heat  conduction,  work  into  heat  of  friction,  expansion  without  work,  and 

diffusion  of  gases 72 

SECTION  C 

Of  negative  change  of  entropy 
Some  of  its  physical  features  and  necessary  accompaniments 78 

SECTION  D 

Physical  significance  of  the  equivalents  for  growth  of  entropy  given  on 

pp.  42-43 80 

SECTION  E 

Physical  significance  of  the  more  specific  statements  of  second  law  given 

on  pp.  44-47 81 

PART  IV 

SUMMARY    OF     THE     CONNECTION     BETWEEN     PROBABILITY,      IRREVERSI- 
BILITY,  ENTROPY,   AND   THE  SECOND  LAW 

SECTION  A 

(i)  Prerequisites    and   conditions    necessary  for   the   application   of  the 
theory  of  probabilities 

(a)  Atomic  theory;  (6)  like  particles;  (c)  very  numerous  particles;  (d) 

11  elementary  chaos  " 83 

(2)  Differences  in  the  states  of  "elementary  chaos" 85 

(3)  Number  of  complexions,  or  probability  of  a  chaotic  state...     86 


X  CONTENTS 

SECTION  B 

PA.QB 

Irreversibility 85 

SECTION  C 
Entropy 87 

SECTION  D 

The  Second  Law 

Its  basis  and  best  statements;  it  has  no  independent  significance 88 

PART  V 

REACH  OR  SCOPE   OF  THE   SECOND  LAW 

SECTION  A 

Its  extension  to  all  bodies 
PLANCK'S  presentation;  fifteen  steps  in  the  proof 91 

SECTION  B 
General  conclusion  as  to  entropy  changes  98 


THE   PHYSICAL  SIGNIFICANCE  OF  ENTROPY 
AND  OF  THE  SECOND  LAW 

[There  is  no  difference  between  change  of  Entropy  and  Second  Law,  when  each 

is  fully  defined.] 


INTRODUCTION 

PURPOSE,  ACKNOWLEDGMENTS,  THE  Two  METHODS  OF  APPROACH 
AND  OUTLINE  OF  TREATMENT 

THIS  article  is  intended  for  those  students  of  engineering  who 
already  have  some  elementary  knowledge  of  thermodynamics. 
It  is  intended  to  clear  up  a  difficulty  that  has  beset  every  earnest 
beginner  of  this  subject.  The  difficulty  is  not  one  of  application 
to  engineering  problems,  although  here  too  there  have  been 
widespread  misconceptions,1  for  the  expressions  developed  by 
CLAUSIUS  are  simple,  have  long  been  known  and  much  used  by 
engineers  and  physicists.  The  difficulty  is  rather  as  to  the 
ultimate  physical  meaning  of  entropy.  This  term  has  long  been  J 
known  as  a  sort  of  property  of  the  state  of  the  body,  has  long  been  ' 
surmised  to  be  of  essentially  a  statistical  nature,  but  with  it  all 
there  was  a  sense  that  it  was  a  sort  of  mathematical  fiction, 
that  it  was  somehow  unreal  and  elusive,  so  it  is  no  wonder 
that  in  certain  engineering  quarters  it  was  dubbed  the  "ghostly 
quantity." 

Now  this  instinct  of  the  true  engineer  to  understand  things 

1  See   Entropy,  by   JAMES  SWINBURNE;  this  author   has  called   attention  to 
necessary  corrections  and  duly  emphasized  the  engineering  aspect. 


2  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

down  to  the  bottom  is  worthy  of  all  encouragement  and  respect. 
For  this  reason  and  because  the  matter  is  of  prime  importance 
to  the  technical  world,  the  final  .meaning  of  entropy  (i.e.,  of  the 
Second  Law)  must  be  -clarified  and  realized.  Indeed,  we  may  well 
go  beyond  this  somewhat  narrow  view  and  say  that  this  is  well 
worth  doing  because  change  of  entropy  constitutes  the  driving  motive 
,in  all  natural  events;  it  has  therefore  a  reach  and  a  universality 
which  even  transcends  that  of  the  First  Law,  or  Principle-of  the 
^Conservation  of  Energy. 

In  striving  to  present  the  physical  meaning  of  entropy  and  of  the 
Second  Law,  the  writer  cannot  lay  claim  to  any  originality;  he 
has  simply  tried  here  to  put  in  logical  order  the  somewhat  scattered 
propositions  of  the  leading  investigators  of  this  subject  and  in 
such  a  way  that  the  difficulties  of  apprehension  might  be  minim- 
ized; in  other  words,  to  present  the  solutions  of  his  own  diffi- 
culties, in  the  hope  that  the  solutions  may  be  helpful  to  other 
students  of  engineering  and  thermodynamics.  In  overcoming  these 
difficulties,  the  writer  owes  everything  to  the  books  and  papers  by 
PLANCK  and  BOLTZMANN,  pre-eminently  to  PLANCK,  who  has  so 
clearly  and  appreciatively  interpreted  the  life  work  of  BOLTZ- 
MANN. l  The  writer  furthermore  wishes  to  say  that  he  has  not 
hesitated  here  to  quote  verbatim  from  both  these  investigators 
and  not  always  so  that  their  own  statements  can  be  distinguished 
from  his  own.  If  any  part  of  this  presentation  is  particularly 
clear  and  exact  the  reader  will  be  safe  in  crediting  it  to  one  or 
the  other  of  these  two  investigators  and  expositors,  although  it 
would  not  be  right  to  consider  them  responsible  for  everything 
contained  in  this  little  book. 

In  considering  the  proper  approach  to  the  matter  in  hand  we 
must  remember  that2  "in  physical  science  there  are  two  more  or 

1  BOLTZMANN,  Gas  Theorie;  PLANCK,  Thermodynamik,  Theorie  der  Warme- 
strahlung,  and  Acht  Vorlesungen  fiber  Theoretische  Physik. 

2  Professor  W.  S.  FRANKLIN,  The  Second  Law  of  Thermodynamics:  its  basis 
in  Intuition  and  Common  Sense.  Pop.  Science  Monthly,  March,  1910. 


AND    OF   THE  SECOND  LAW  3 

less  distinct  modes  of  attack,  namely,  (a)  a  mode  of  attack  in 
which  the  effort  is  made  to  develop  conceptions  of  the  physical 
processes  of  nature,  and  (b)  a  mode  of  attack  in  which  the  attempt 
is  made  to  correlate  phenomena  on  the  basis  of  sensible  things, 
things  that  can  be  seen  and  measured.  In  the  theory  of  heat  the 
first  mode  is  represented  by  the  application  of  the  atomic  theory 
to  the  study  of  heat  phenomena,  and  the  second  mode  is  represented 
by  what  is  called  thermodynamics."  In  solving  the  special 
problem  before  us,  as  to  the  physical  meaning  of  entropy  and  of 
the  Second  Law,  our  main  dependence  must  be  on  the  first  mode 
of  attack. 

The  second  mode  will  furnish  checks  and  confirmations  of  the 
results  developed  by  the  first,  or  we  may  say  that  the  combination 
of  the  two  modes  will  give  the  well-established  characteristic 
equations  and  relations  of  bodies  and  their  physical  elements. 

The  whole  discussion  will  now  be  taken  up  in  a  non-mathe- 
matical way,  without  the  full  proof  required  by  a  complete  presenta- 
tion, and  about  in  this  order: 

(a)  The  definitions,  general  preliminaries  and  current  state- 
ments of  the  matters  considered. 

(6)  More  or  less  precise  statement  of  the  primary  relations 
and  theorems. 

(c)  The  physical  interpretations. 

(d)  Summary  of  the  connection  between  probability,  irreversi- 
bility,  entropy  and  the  Second  Law. 

(e)  Reach  or  scope  of  the  Second  Law. 

On  account  of  the  difficulty  which  every  student  experience^in 
realizing  the  physical  nature  of  entropy  we  will  in  the  main  con- 
fine our  attention  here  to  gases  and  indeed  to  their  simplest  case, 
the  monatomic  gas,  and  will  as  usual  assume  that  the  dimen- 
sions of  an  atom  or  particle  are  very  small  in  comparison  with  the 
average  distance  between  two  adjacent  particles,  that  for  the  atoms 
approaching  collision  the  distance  within  which  they  exert  a  signifi- 
cant influence  on  each  other  is  very  small  as  compared  with  the 


4  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

mean  distance  between  adjacent  atoms,  and  that  between  collisions 
the  mean  length  of  the  particle's  path  is  great  in  comparison  with 
the  average  distance  between  the  particles.  Later  on  we  will  indi- 
cate in  a  very  general  and  brief  way  how  the  entropy  idea  may 
be  extended  to  other  states  of  aggregation  and  to  other  than 
purely  thermodynamic  phenomena.  Mostly,  therefore,  we  will 
only  consider  states  and  processes  in  which  heat  phenomena  and 
mechanical  occurrences  take  place. 


AND  OF   THE  SECOND   LAW 


PART  I 

DEFINITIONS,  GENERAL  PRELIMINARIES,  DEVELOPMENT,  CURRENT 
AND  PRECISE  STATEMENTS  OF  THE  MATTERS  CONSIDERED 

(i)  The  "  State  "  of  a  Body  and  its  "  Change  of  State  " 

As  we  will  make  constant  use  of  the  terms  contained  in  this 
heading  and  as  they  here  represent  fundamentally  important 
conceptions,  we  will  seek  to  make  them  clear  by  presenting  them 
in  the  various  forms  into  which  they  have  been  cast  by  the  different 
investigators,  even  at  the  risk  of  being  considered  prolix. 

In  the  Introduction  to  this  article  we  called  attention  to  the 
two  distinct  modes  of  attacking  any  physical  problem.  Now  the 
conception  "state  of  a  body"  varies  with  the  chosen  mode  of 
attack.  Of  course  as  both  modes  are  legitimate  and  lead  to 
correct  results,  these  differences  in  the  conception  of  "state" 
can  be  reconciled  and  a  broader  definition  reached.  We  can 
illustrate  these  different  methods  of  approach,  as  PLANCK  has 
done,  by  assuming  two  different  observers  of  the  state  of  the  body, 
one  called  the  microscopic-observer  and  the  other  the  macro- 
scopic-observer. The  former  possesses  senses  so  acute  and 
powers  so  great  that  he  can  recognize  each  individual  atom  and 
can  measure  its  motion.  For  this  observer  each  atom  will  move 
exactly  according  to  the  elementary  laws  prescribed  for  it  by 
General  Dynamics.  These  laws,  so  far  as  we  know  them,  also 
at  once  permit  of  exactly  the  opposite  course  of  each  event.  Con- 
sequently there  can  be  here  no  question  of  probability,  of  entropy 
or  of  its  growth.  On  the  other  hand,  the  "macro-observer,"  (who 
perceives  the  atomic  host,  say  as  a  homogeneous  gas,  and  conse- 


6  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

quently  applies  to  its  mechanical  and  thermal  events  the  laws 
of  thermodynamics)  will  regard  the  proc3S3  as  a  whole  to  be 
an  irreversible  one  in  accordance  with  the  Second  Law.  .  .  . 
Now  a  particular  change  of  state  cannot  at  the  same  time  be 
both  reversible  and  irreversible.  But  the  one  observer  has  a 
different  idea  of  "  change  of  state"  from  the  other;  the  micro- 
observer's  conception  of  "  change  of  state "  is  different  from 
that  of  the  macro-observer.  What  then  is  " change  of  state?" 
The  state  of  a  physical  system  can  probably  not  be  rigorously 
denned,  otherwise  than  the  conception,  as  a  whole,  of  all  those 
physical  magnitudes  whose  instantaneous  values,  under  given 
external  conditions,  also  uniquely  determine  the  sequence  of  these 
changing  values. 

BOLTZMANN'S  statement  is  much  more  clear,  namely,  "The 
state  of  a  body  is  determined,  (a)  by  the  law  of  distribution  of 
the  particles  in  space  and  (b)  by  the  law  of  distribution  of 
the  velocities  of  the  particles;  in  other  words,  a  body's  condition 
is  determined  (a)  by  the  number  of  particles  which  lie  in  each 
elementary  realm  of  the  space  and  (b)  by  a  statement  of  the 
number  of  particles  which  belong  to  each  elementary  velocity 
group.  These  elementary  realms  are  all  equal  and  so  are  the  ele- 
mentary velocity  groups  equal  among  themselves.  But  it  is  further- 
more assumed  that  each  elementary  realm  and  each  elementary 
velocity  group  contains  very  many  particles." 

Now  if  we  ask  the  aforesaid  two  observers  what  they  under- 
stand by  the  state  of  the  atomic  host  or  gas  under  consideration, 
they  will  give  entirely  different  answers.  The  micro-observer 
will  mention  those  magnitudes  which  determine  the  location  and 
the  velocity  condition  of  all  the  individual  atoms.  This  would 
mean  in  the  simplest  case,  in  which  the  atoms  are  regarded  as 
material  points,  that  there  would  be  six  times  as  many  magnitudes 
as  atoms  present,  namely,  for  each  atom  there  would  be  three 
co-ordinates  of  location  and  three  of  velocity  components;  in 
the  case  of  composite  molecules  there  would  be  many  more  such 


AND   OF  THE  SECOND   LAW  7 

magnitudes.  For  the  micro-observer,  the  state  and  the  sequence 
of  the  event  would  not  be  determined  until  all  these  many  mag- 
nitudes had  been  separately  given.  The  state  thus  defined  we 
will  call  the  "micro-state."  The  macroscopic-observer  on  the 
other  hand  gets  along  with  much  fewer  data;  he  will  say  that  the 
state  of  the  contemplated  homogeneous  gas  is  already  determined 
by  the  density,  the  visible  velocity  and  the  temperature  at  each 
place  of  the  gas  and  he  will  expect,  when  these  magnitudes  are 
given,  that  the  course  of  the  physical  events  will  be  completely 
determined,  namely,  will  occur  in  obedience  to  the  two  laws  of 
thermodynamics  and  therefore  be  bound  to  show  an  increase  in 
entropy.  The  state  thus  denned  we  will  call  the  "macro-state." 
The  difference  in  the  two  observers  is  that  one  sees  only  the  atomic 
events  and  the  other  the  occurences  in  the  aggregate.  The  former 
would  have  the  absolute  mechanical  idea  of  state  and  the  latter 
the  statistical  idea.  Before  attempting  to  reconcile  their  apparently 
conflicting  conclusions,  we  will  here  call  attention  to  some  neces- 
sary relations  between  the  micro-state  and  the  macro-state. 
In  the  first  place  we  must  remember  that  all  a  priori  possible 
micro-states  are  not  realized  in  nature;  they  are  conceivable 
but  never  attain  fruition.  How  shall  we  select  what  may  be 
called  these  natural  micro-states?  The  principles  of  general 
dynamics  furnish  no  guide  for  such  selection  and  so  recourse 
may  be  had  to  any  dynamic  hypothesis  whose  selection  will  be 
fully  justified  by  experience. 

Now  PLANCK  says:  "  In  order  to  traverse  this  path  of  investi- 
gation, we  must  evidently  first  of  all  keep  in  mind  all  the  con- 
ceivable positions  and  velocities  of  the  individual  atoms,  which 
are  compatible  with  particular  values  of  the  density,  the  velocity 
and  the  temperature  of  the  gas,  or,  in  other  words,  we  must  con- 
sider all  the  micro-states  which  belong  to  a  particular  macro- 
state  and  must  examine  all  the  different  events  which  follow 
from  the  different  micro-states  according  to  the  fixed  laws  of 
dynamics.  Now  up  to  this  time,  the  closer  calculation  and  com- 


8  THE   PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

bination  of  these  minute  elements  has  always  given  the  important 
result  that  the  vast  majority  of  these  micro-states  belong  to  one 
and  the  same  macro-state  or  aggregate,  and  that  only  compara- 
tively few  of  the  said  micro-states  furnish  an  anomalous  result, 
and  these  few  are  characterized  by  very  special  and  far-reaching 
conditions  existing  between  the  locations  and  the  velocities  of 
adjacent  atoms.  And,  furthermore,  it  has  appeared  that  the 
almost  invariably  resulting  macro-event  is  just  the  very  one 
perceived  by  the  macroscopic  observer,  the  one  in  which  all 
the  measureable  mean  values  have  a  unique  sequence,  and  con- 
sequently and  in  particular  satisfies  the  second  law  of  thermody- 
namics." 

"  Herewith  is  revealed  the  bridge  of  reconciliation  between 
the  two  observers.  The  micro- observer  needs  only  to  take  up 
in  his  theory  the  physical  hypothesis,  that  all  such  particular 
cases  (which  premise  very  special,  far-reaching  conditions  between 
the  states  of  adjacent  and  interacting  atoms)  do  not  occur  in 
Nature;  or  in  other  words,  the  micro-states  are  in  '  elementary 
disorder '  (elementar  ungeordnet).  This  secures  the  unique 
(unambiguous)  character  of  the  macroscopic  event  and  makes 
sure  that  the  Principle  of  the  Growth  of  Entropy  will  be  satisfied 
in  every  direction." 

Before  elaborating  all  that  is  implied  in  this  hypothesis  of 
"  elementary  disorder  "  we  will  again  point  out  that  for  each 
macro-state  (even  with  settled  values  of  density  and  temperature) 
there  may  be  many  micro-states  which  satisfy  it  in  the  aggregate. 
According  to  PLANCK,  "  it  is  easy  to  see  that  the  macro-observer 
deals  with  mean  values;  for  what  he  calls  density,  visible  velocity, 
temrjerature  of  the  gas,  are  for  the  micro-observer  certain  averages, 
statistical  data,  which  have  been  suitably  obtained  from  the 
spatial  arrangement  and  the  velocities  of  the  atoms.  But  with 
these  averages  the  micro-observer  at  first  can  do  nothing  even 
if  they  are  known  for  a  certain  time,  for  thereby  the  sequence 
of  events  is  by  no  means  settled;  on  the  contrary,  he  can  easily 


AND   OF  THE  SECOND  LAW  9 

with  said  given  averages  ascertain  a  whole  host  of  different  values 
for  the  location  and  velocities  of  the  individual  atoms,  all  of  which 
correspond  to  said  given  averages,  and  yet  some  of  these  lead  to 
wholly  different  sequences  of  events  even  in  their  mean  values," 
events  which  do  not  at  all  accord  with  experience.  It  is  evident, 
if  any  progress  is  to  be  made,  that  the  micro-observer  must  in 
some  suitable  way  limit  the  manifold  character  of  the  multi- 
farious micro-states.  This  he  accomplishes  by  the  hypothesis 
of  "  elementary  disorder  "  about  to  be  more  fully  denned. 

In  passing  we  may  here  note  for  future  use,  that  what  has 
just  been  said  concerning  macro-states  (aggregates)  with  "  settled  " 
mean  velocity,  density  and  temperature,  applies  also  to  states 
unsettled  in  the  aggregate,  so  far  as  concerns  the  manifold  char- 
acter of  the  conceivable  constituent  micro-states  and  the  dif- 
ferences in  the  mean  character  of  their  sequences.  Even  after 
the  above  limiting  hypothesis  removes  all  illegitimate  micro- 
states,  an  enormously  greater  number  of  legitimate  ones  will  be 
left  to  constitute  the  number  of  complexions  properly  belonging 
to  the  state  contemplated.  We  may  also  add  that  it  seems  quite 
evident  that  the  numbers  representing  these  complexions  will 
be  different  in  the  settled  and  unsettled  states  even  if  the  latter 
should  ultimately  possess  the  mean  velocity,  density  and  tem- 
perature of  the  former. 

On  the  other  hand,  we  also  point  out  that  for  one  and  the  same 
set  of  external  conditions  the  macro-state  may  itself  vary  very 
greatly.  When  it  has  a  settled  density  and  temperature,  it  is 
said  to  be  in  a  stationary  state,  to  be  in  thermal  equilibrium  and, 
anticipating,  we  may  add  that  it  is  then  has  maximum  entropy, 
in  short  we  may  say  it  is  in  a  "  normal  "  condition.  But  the 
external  conditions  remaining  the  same,  before  attaining  to 
said  "  normal  "  ultimate  state,  it  may  pass  through  a  whole 
series  of  so-called  "  abnormal  "  states  after  it  leaves  its  initial 
condition.  While  it  is  in  any  one  of  these  "  abnormal  "  states, 
it  may  be  said  to  be  in  a  more  or  less  turbulent  condition 5 


10  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

it  may  then  possess  whirls  and  eddies;  it  may  have  different 
densities  and  temperatures  in  its  different  parts  and  then  it  will 
be  difficult  or  impossible  to  measure  these  external  physical 
features  of  its  state  as  a  whole.  All  this  implies  ever-varying 
atomic  locations  and  velocities,  but  does  not  indicate  any  such 
special  far-reaching  regularities  between  adjacent  and  inter- 
acting particles  as  would  vitiate  at  any  stage  our  hypothesis  of 
"  elementary  disorder "  (elementar  ungeordnet)  or  "  molecular 
chaos." 

Before  going  into  more  detail  concerning  this  particular  chaotic 
condition  of  the  particles  we  will  give  PLANCK'S  somewhat  fuller 
statement  of  what  constitutes  the  "  state  "  of  a  physical  system 
at  a  particular  time  and  under  given  external  conditions.  It  is, 
"  the  conception  as  a  whole  of  all  those  mutually  independent 
magnitudes  which  determine  the  sequence  of  events  occurring 
in  the  system  so  far  as  they  are  accessible  to  measurement;  the 
knowledge  of  the  state  is  therefore  equivalent  to  a  knowledge 
of  the  initial  conditions.  For  example,  in  a  gas  composed  of 
invariable  molecules  the  state  is  determined  by  the  law  of  their 
space  and  velocity  distribution,  i.e.,  by  the  statement  of  the 
number  of  molecules,  of  their  co-ordinates  and  velocity  compo- 
nents which  lie  within  each  single  small  region.  The  number 
of  molecules  in  any  one  of  these  different  regions  is  in  general 
entirely  independent  of  the  number  in  any  other  region,  for  the 
state  need  not  be  a  stationary  one  nor  one  of  equilibrium;  these 
numbers  should  therefore  all  be  separately  known  if  the  state 
of  the  gas  is  to  tie  considered  as  given  in  the  absolute  mechanical 
sense.  On  the  other  hand,  for  the  characterization  of  the  state 
in  the  statistical  sense,  it  is  not  necessary  to  go  into  closer  detail 
concerning  the  molecules  present  in  each  elementary  space; 
for  here  the  necessary  supplement  is  supplied  by  the  hypothesis 
of  molecular  chaos,  "  which  in  spite  of  its  mechanically  indeter- 
minate character  guarantees  the  unambiguous  sequence  of  the 
physical  events." 


AND  OF  THE  SECOND   LAW  11 


(2)  Further  Elucidation  of  this  Essential  Condition  of  "  Elemen- 
tary Chaos. ' '     Sundry  A  spects  of  Haphazard 

To  gain  as  complete  an  understanding  as  possible  of  this  funda- 
mental idea  we  will  now  give  the  views  of  the  several  investigators 
as  to  the  physical  features  of  this  chaotic  state.  We  have  seen 
how  PLANCK,  the  chief  expositor  of  BOLTZMANN,  boldly  excludes 
from  consideration  all  cases  leading  to  anomalous  results,  because 
of  the  very  special  conditions  existing  between  the  molecular  data, 
by  assuming  that  these  cases  do  not  occur  in  Nature.  PLANCK 
reminds  the  physicists  who  object  to  the  hypothesis  of  elementary 
disorder  because  they  feel  it  is  unnecessary  or  even  unjustifiable, 
that  the  hypothesis  is  already  much  used  in  Physics,  that  tacitly 
or  otherwise  it  underlies  every  computation  of  the  constants 
attached  to  friction,  diffusion  and  the  conduction  of  heat.  On 
the  other  hand  he  reminds  others,  those  inclined  to  regard  the 
hypothesis  of  "  elementary  disorder  "  as  axiomatic,  of  the  theorem 
of  H.  POINCARE,  which  excludes  this  hypothesis  for  all  times  from 
a  space  surrounded  with  absolutely  smooth  walls.  PLANCK  says 
that  the  only  escape  from  the  portentous  sweep  of  this  proposition 
is  that  absolutely  smooth  walls  do  not  exist  in  Nature. 

The  foregoing  thought  PLANCK  has  also  put  in  a  slightly  dif- 
ferent way.  Appreciating  that  all  mechanically  possible  simul- 
taneous arrangements  and  velocities  of  molecules  are  not  realized 
in  Nature,  the  concept  of  "  elementary  diforder  "  implies  one 
limitation  of  the  conceivable  molecular  states,  namely  that, 
between  the  numerous  elements  of  a  physical  system  there  exist 
no  other  relations  than  those  conditioned  by  the  existing  measurable 
mean  values  of  the  physical  features  of  the  system  in  question. 

Another,  briefer  but  equivalent,  definition  is  that:  "  In  Nature 
all  states  and  processes  which  contain  numerous  independent 
(unkontrollierbar)  constituents  are  in  '  elementary  disorder ' 
(elementar  ungeordnet) ."  The  constituents  are  molecular  ele- 


12  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

ments  in  mechanics  and  in  thermodynamics  and  the  energy 
elements  in  radiation. 

The  German  word  "  unkontr oilier bar  ?>1  here  used  may  also 
with  some  justice  be  translated  as,  unconditioned,  undetermined, 
unmeasurable,  unregulated,  uncorrelated,  ungovernable  or  hap- 
hazard. But  whichever  term  is  best,  PLANCK,  mechankally 
srjeakmg^jneant  by  it,  the  confused,  unregulated  and  whirring 
intermingling  of  very  many  atoms. 

Either  of  these  two  equivalent  definitions  implies  that  such 
elementary  disorder  or  chaos  is  a  condition  of  sufficiently  com- 
plete haphazard  to  warrant  the  application  of  the  .Theory.,  of 
Probabilities  to  the  unique  (unambiguous)  determination  joL  the 
measurable  physical  features  of  the  process  viewed  as  a  whole. 

The  foregoing  ideas  more  or  less  tacitly  underlie  the  whole 
of  B^OLTZM ANN'S  great  pioneer  work  in  this  vast  field.  He  it  was 
who  clearly  showed  that  the  Second  Law  could  be  derived  from 
mechanical  principles:  that  entropy  was  a  property  of  every 
state,  turbulent  or  otherwise;  that  the  entropy  idea  would  be 
emancipated  from  all  thought  of  human,  experimental,  skill, 
and  who  thereby  raised  the  Second  Law  to  the  position  of  a  real 
principle.  He  did  all  this  by  a  general  basing  of  the  idea  of  entropy 
on  the  idea  of  probability.  Consequently  we  find  much  attention 
paid  in  all  his  work  to  haphazard  molecular  conditions.  He 
first  used  the  terms  "  molekular-geordnet  "  (molecularly  ordered, 
or  arranged),  and  "  molekular  ungeordnet  "  ^  molecular  ly  dis- 
ordered or  disarranged),  which  latter  phrase  we  must  regard 
as  synonomous  with  the  term  "  ekmentar  ungeordnet "  (elemen- 
tary disorder  or  chaos)  with  which  we  have  already  become 
acquainted  in  PLANCK'S  presentation.  We  will,  therefore,  con- 
fine ourselves  here  to  BOLTZMANN'S  illustrations  of  these  terms, 
for  his  work  does  not,  in  these  particulars,  contain  any  sharp 


1  On  p.  133  of  Warmestrahlung  PLANCK  says,  "only  measurable  mean  values 
are  kontrollierbar, "  and  this  may  help  us  to  get  the  meaning  here. 


AND  OF  THE  SECOND  LAW  13 

definitions.  Indeed  he  may  have  feared  over-precision  and  may 
have  trusted  to  the  use  he  made  of  the  terms  at  different  tunes 
to  convey  their  meaning. 

Concerning  some  of  the  characteristics  of  BOLTZM ANN'S  hap- 
hazard motion  we  take  the  following  from  Vol.  I  of  his  "  Vorle- 
sungen  liber  Gas  Theorie." 

If  in  a  finite  part  of.  a  gas  the  variables  determining  the  motion 
of  the  molecules  have  different  mean  values  from  those  in  another 
finite  part  of  the  gas  (for  example  if  the  mean  density  or  mean 
velocity  of  a  gas  in  one-half  of  a  vessel  is  different  from  those  in 
the  other  half),  or  more  generally,  if  any  finite  part  of  a  gas  behaves 
differently  from  another  finite  part  of  a  gas,  then  such  a  dis- 
tribution is  said  to  be  "  molar-geordnet "  (in  molar  order). 
But  when  the  total  number  of  molecules  in  every  unit  of  volume 
exists  under  the  same  conditions  and  possesses  the  same  number 
of  each  kind  of  molecules  throughout  the  changes  contemplated, 
then  the  same  number  of  molecules  will  leave  a  unit  volume  and 
will  enter  it  so  that  the  total  number  ever  present  remains  the  same; 
under  such  conditions  we  call  the  distribution  "  molar-ungeordnet " 
(in  molar  disorder)  and  that  finite  distribution  is  one  of  the 
characteristics  of  the  haphazard  state  to  which  the  Theory  of 
Probabilities  is  applicable.  [As  another  illustration  of  the  excluded 
molar-geordnet  states  we  may  instance  the  case  when  all  motions 
are  parallel  to  one  plane.] 

But  although  in  passing  from  one  finite  part  to  another  of  a  gas 
no  regularities  (of  average  character)  can  be  discerned,  yet  infin- 
itesimal parts  (say  of  two  or  more  molecules)  may  exhibit  certain 
regularities,  and  then  the  distribution  would  be  "  molekular- 
geordnet "  (molecularly-ordered)  although  as  a  whole  the  gas 
is  "  molar-ungeordnet."  For  example  (to  take  one  of  the  infinite 
number  of  possible  cases)  suppose  that  the  two  nearest  mole- 
cules always  approached  each  other  along  their  line  of  centers, 
or  if  a  molecule  moving  with  a  particularly  slow  speed  always 
had  ten  (10)  slow  neighbors,  then  the  distribution  would  be 


14  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

"  molekular-geordnet."  But  then  the  locality  of  one  molecule 
would  have  some  influence  on  the  locality  of  another  molecule 
and  then  in  the  Theory  of  Probabilities  the  presence  of  one  mole- 
cule in  one  place  would  not  be  independent  of  the  presence  of 
some  other  molecule  in  some  other  place.  Such  dependence 
is  not  permissible  by  the  Theory  of  Probabilities.  Before,  how- 
ever, we  can  further  describe  what  is  here  perhaps  the  most  impor- 
tant term  (molekular-ungeordnet),  we  must  point  out  that  BOLTZ- 
MANN  considers  the  number  of  molecules  m  of  one  kind  whose 
component  velocities  along  the  co-ordinate  axes  are  confined 
between  the  limits, 


£  and  £+d£,     t)  and  y+dr),     £  and  d^,       .     .     (i) 

and  also  the  number  of  molecules  mi  of  another  kind  whose 
component  velocities  similarly  lie  between  the  limit 

£i*£i  +  £i,     gi*fli+<fyi,     Si  •*-«!,      ...     (2) 

then,  considering  the  chances  that  a  molecule  m  shall  have 
velocities  between  the  limits  specified  in  (i)  and  molecule  mi 
have  velocities  between  limits  (2),  BOLTZMANN  intimates  that  these 
chances  are  independent  of  the  relative  position  of  the  molecules. 
Where  there  is  such  complete  independence,  or  absence  of  all 
minute  regularities,  the  distribution,  according  to  BOLTZMANN, 
is  "  molekular-ungeordnet  "  (molecular  ly-disordered). 

BOLTZMANN  furthermore  informs  us  that,  as  soon  as  in  a  gas, 
the  mean  length  of  path  is  great  in  comparison  with  the  mean  dis- 
tance between  two  adjacent  molecules,  the  neighboring  molecules 
will  quickly  become  different  from  what  they  formerly  were. 
Therefore  it  is  exceedingly  probable  that  a  "  molekular-geord- 
nete  "  (but  molar-ungeordnete)  distribution  would  shortly  pass 
into  a  "  molekular-ungeordnete  "  distribution. 

Furthermore,  from  the  constitution  of  a  gas  results  that  the 
place  where  a  molecule  collided  is  entirely  independent  of  the 
spot  where  its  preceding  collision  took  place.  Of  course,  this 


AND  OF  THE  SECOND  LAW  15 

independence  could  be  maintained  for  an  indefinite  time  only 
by  an  infinite  number  of  molecules. 

The  place  of  collision  of  a  pair  of  molecules  must  in  our 
Theory  of  Probabilities  be  independent  of  the  locality  from 
which  either  molecule  started. 

From  all  the  preceding  we  must  infer  what  measure  of  hap- 
hazard BOLTZMANN  considers  necessary  for  the  legitimate  use  of 
the  Theory  of  Probabilities. 

BOLTZMANN  in  proving  his  H-Theorem,1  which  establishes  the 
one-sidedness  of  all  natural  events,  makes  the  explicit  assumption 
that  the  motion  at  the  start  is  both  "  molar-  und  molekular- 
ungeordnet "  and  remains  so.  Later  on,  he  assumes  the  same 
things  but  adds  that  if  they  are  not  so  at  the  start  they  will  soon 
become  so;  therefore  said  assumption  does  not  preclude  the  con- 
sideration by  Probability  methods  of  the  general  case  or  the 
passage  from  "  ordnete  "  to  "  ungeordnete  "  conditions  which 
characterizes  all  natural  events. 

In  fact  these  very  definitions  show  solicitude  for  securing  the 
uninterrupted  operation  of  the  laws  of  probability.  BOLTZMANN 
intimates  his  approval  of  S.  H.  BURBURY'S  statement  of  the  con- 
dition of  independence  underlying  his  work. 

Here  S.  H.  BURBURY  2  simplifies  the  matter  by  assuming  that 
any  unit  of  volume  of  space  contains  a  uniform  mixture  of 
differently  speeded  molecules  and  then  says: 

"  Let  V  be  the  velocity  of  the  center  of  gravity  of  any  pair  of 
molecules  and  R  their  relative  velocity.  Then  the  following 
condition  (here  called  A)  holds:  For  any  given  direction  of  R 
before  collision,  all  directions  of  R  after  collision  are  equally 
probable.  Then  BOLTZMANN'S  H-theorem  proves  that  if  con- 
dition A  be  satisfied,  then  if  all  directions  of  the  relative  velocity 
R  for  given  V  are  not  equally  likely,  the  effect  of  collisions 

1  In  BOLTZMANN'S  H-Theorem  we  have  a  process  (consisting  of  a  number  of 
separately  reversible  processes)  which  is  irreversible  in  the  aggregate. 

2  Nature,  Vol.  LI,  p.  78,  Nov.  22,  1894. 


16  THE  PHYSICAL   SIGNIFICANCE  OF  ENTROPY 

is  to  make  H  diminish."  [In  essence  BURBURY'S  condition 
A  says  no  more  than  that  Theory  of  Probabilities  is  applicable 
for  finding  number  of  collisions.]  Furthermore,  "  any  actual 
material  system  receives  disturbances  from  without,  the  effect 
of  which  coming  at  haphazard  without  regard  to  state  of 
system  for  the  time  being  is,  pro  tanto,  to  renew  or  maintain 
the  independence  of  the  molecular  motions,  that  very  distribu- 
tion of  co-ordinates  (of  collision)  which  is  required  to  make 
H  diminish.  So  there  is  a  general  tendency  for  H  to  diminish, 
though  it  may  conceivably  increase  in  particular  cases.  Just 
as  in  matters  political,  change  for  the  better  is  possible,  but  the 
tendency  is  for  all  change  to  be  from  bad  to  worse."  Here 
BURBURY  states  what  is  practically  true  in  all  actual  cases  and 
thus  furnishes  an  additional  reason,  if  that  were  needed,  for  the 
legitimacy  of  the  Probability  method  pursued  by  BOLTZMANN,  and, 
another  explanation  of  why  the  results  obtained  are  in  such  per- 
fect accord  with  experience. 

AsBuRBURY'sremarkswith  respect  to  the  nature  of  "elementary 
chaos "  under  consideration  are  always  illuminating,  we  will, 
at  the  risk  of  repeating  something  already  said,  quote  the  following: 

"  The  chance  that  the  spheres  approaching  collision  shall 
have  velocities  within  assigned  limits  is  independent  of  their 
relative  position,  and  of  the  positions  and  velocities  of  all  other 
spheres,  and  also  independent  of  the  past  history  of  the  system 
except  so  far  as  this  has  altered  the  distribution  of  the  velocities 
inter  se.  In  the  following  example  this  independence  is  satisfied 
for  the  initial  state  and,  for  the  assumed  method  of  distribution, 
has  no  past  history. 

"  Example.  A  great  number  of  equal  elastic  spheres,  each  of 
unit  mass  and  diameter  a,  are  at  an  initial  instant  set  in  motion 
within  a  field  S  of  no  force  and  bounded  by  elastic  walls.  The 
initial  motion  is  formed  as  follows:  (i)  One  person  assigns  com- 
ponent velocities  «,  v,  w  to  each  sphere  according  to  any  law 
subject  to  the  conditions  that  2u=I>v=I,w=Q  and  that 


AND   OF   THE  SECOND  LAW  17 

a  given  constant.  (2)  Another  person,  in  com- 
plete ignorance  of  the  velocities  so  assigned,  scatters  the  spheres 
at  haphazard  throughout  S.  And  they  start  from  the  initial 
positions  so  assigned  by  (2)  with  the  velocities  assigned  to  them 
respectively  by  (i)." 

The  system  thus  synthetically  constructed  would  without 
doubt,  at  the  start  be  "  molekular-ungeordnet  " — in  fact,  it  is  as 
near  an  approach  to  chaos  as  is  possible  in  an  imperfect  world. 
But  there  is  reason  to  doubt  if  it  would  continue  to  be  thus  "  molek- 
ular-ungeordnet." For  the  distribution  of  velocities  is  according 
to  any  law  consistent  with  the  above-mentioned  conditions  and 
some  such  laws  would  lead  to  results  hostile  to  the  Second  Law, 
and  then  we  may  safely  say  such  laws  of  velocity  distribution 
would  never  occur  in  Nature  and  would  therefore  belong  to  the 
cases  which  have  been  specially  excepted. 

Now  there  are  mechanical  systems  which  possess  the  entropy 
property  and  it  has  been  truly  said  that  the  Second  Law  and  irre- 
versibility  do  not  depend  on  any  special  peculiarity  of  heat  motion, 
but  only  on  the  statistical  property  of  a  system  possessing  an 
extraordinary  number  of  degrees  of  freedom.  In  this  sense 
Professor  J.  W.  GIBBS  treated  Mechanics  statistically  and  showed 
that  then  the  properties  of  temperature  and  entropy  resulted. 
This  matter  has  already  been  touched  upon,  but  as  numerous 
degrees  of  freedom  is  a  feature  of  the  "  elementary  chaos  "  under 
consideration  it  deserves  repetition  here  and  more  than  a  passing 
mention. 

Illustration  of  Degrees  of  Freedom.  Refer  a  body's  motion  to 
three  axes,  X,  Y,  Z.  If  a  body  has  as  general  a  motion  as  possible, 
it  may  be  resolved  into  translations  parallel  to  the  X,  F,  Z  axes 
and  to  rotations  about  these  axes.  Each  of  these  two  sets  furnishes 
three  components  of  motion  or  a  total  of  six  components;  then 
we  say  that  the  perfectly  unconstrained  motion  of  the  body  has 
six  degrees  of  freedom.  If  a  body  moves  parallel  to  one  of  the 
co-ordinate  planes,  we  say  it  has  two  degrees  of  freedom.  When 


18  THE  PHYSICAL  SIGNIFICANCE   OF  ENTROPY 

we  come  to  consider  molecular  motion  in  general  and  the  inde- 
pendence which  characterizes  the  motion  of  each  of  the  many 
molecules  we  see  that  altogether  we  have  here  an  extraordinary 
number  of  degrees  of  freedom,  and  composed  of  such  is  the  realm 
of  our  "  elementary  chaos." 

If  we  go  to  the  other  extreme  and  think  of  only  one  atom, 
we  see  at  once  that  we  cannot  properly  speak  of  its  disorder. 
But  the  case  is  different  with  a  moderate  number  of  atoms,  say, 
a  hundred  or  a  thousand.  Here  we  surely  can  speak  of  disorder 
if  the  co-ordinates  of  location  and  the  velocity  components  are 
distributed  by  haphazard  among  the  atoms.  But  as  the  process 
as  a  whole,  the  sequence  of  events  in  the  aggregate,  may  not 
with  this  comparatively  small  number  of  atoms  take  place  before 
a  macroscopic  observer  in  a  unique  (unambiguous)  manner, 
we  cannot  say  that  we  have  here  reached  a  true  state  of  "  elemen- 
tary chaos."  If  we  now  ask  as  to  the  minimum  number  of  atoms 
necessary  to  make  the  process  an  irreversible  one,  the  answer  is, 
as  many  as  are  necessary  to  form  determinate  mean  values  which 
will  define  the  progress  of  the  state  in  the  macroscopic  sense. 
Only  for  these  mean  values  does  the  Second  Law  possess  signifi- 
cance; for  these,  however,  it  is  perfectly  exact,  just  as  exact  as 
the  theorem  of  probability,  which  says  that  the  mean  value  of 
numerous  throws  with  one  cubical  die  is  equal  to  3^. 

We  may  now  properly  infer  from  all  these  views  that  the 
state  of  "  elementary  chaos "  (or  "  molekular  ungeordnete  " 
motion)  is  the  necessary  condition  for  adequate  haphazard  and 
makes  the  application  of  the  Theory  of  Probabilities  possible. 

(3)  Settled  and  Unsettled  States;  Distinction  between  Final  Stage 
of  Elementary  Chaos  and  its  Preceding  Stages 

The  immediate  purpose  in  the  next  few  pages  is  to  establish 
the  (a)  distinction  between  the  successive  stages  of  "  elementary 
disorder  "  (chaos)  as  they  develop  in  their  inevitable  passage 


AND  OF   THE  SECOND  LAW  19 

from  "  abnormal  "  conditions  to  the  final  and  so-called  "  normal  " 
condition  of  thermal  equilibrium  and,  furthermore,  (b)  to  show  that 
each  of  these  stages  is  "  elementar-ungeordnet "  and  (c)  that  in 
each  one  sufficient  haphazard  prevails  to  permit  the  legitimate 
application  of  the  Theory  of  Probabilities. 

We  will  first  describe  the  unsettled  (abnormal)  and  settled 
(normal)  states,  respectively.  When  we  consider  the  general 
state  of  a  gas  "  we  need  not  think  of  the  state  of  equilibrium, 
for  this  is  still  further  characterized  by  the  condition  that  its 
entropy  is  a  maximum.  Hence  in  the  general  or  unsettled  state 
of  the  gas  an  unequal  distribution  of  density  may  prevail,  any 
number  of  arbitrarily  different  streams  (whirls  and  eddies)  may 
be  present,  and  we  may  in  particular  assume  that  there  has  taken 
place  no  sort  of  equalization  between  the  different  velocities  of  the 
molecules.  We  may  assume  beforehand,  in  perfectly  arbitrary 
fashion,  the  velocities  of  the  molecules  as  well  as  their  co-ordinates 
of  location.  But  there  must  exist  (in  order  that  we  may  know 
the  state  in  the  macroscopic  sense) ,  certain  mean  values  of  density 
and  velocity,  for  it  is  through  these  very  mean  values  that  the 
state  is  characterized  from  the  macroscopic  standpoint."  The 
differences  that  do  exist  in  the  successive  stages  of  disorder  of  the 
the  unsettled  state  are  mainly  due  to  the  molecular  collisions 
that  are  constantly  taking  place  and  which  thus  change  the 
locus  and  velocity  of  each  molecule. 

We  may  now  easily  describe  the  settled  state  as  a  special  case 
of  the  unsettled  one.  In  the  settled  state  there  is  an  equal  dis- 
tribution of  density  throughout  all  the  elementary  spaces,  there 
are  no  different  streams  (whirls  or  eddies)  present,  and  an  equal 
partition  of  energy  exists  for  all  the  elementary  spaces.  For  it 
tKermal  equilibrium  exists,  the  entropy  is  a  maximum,  and  tem- 
perature of  the  state  has  now  a  definite  meaning,  because  tem- 
perature is  the  mean  energy  of  the  molecules  for  this  state  of 
equilibrium.  The  condition  is  said  to  be  a  "  stationary "  or 
permanent  one,  for  the  mean  values  of  the  density,  velocity,  and 


20  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

temperature    of   this   particular    aggregate    no    longer    change, 
although  molecular  collisions  are  still  constantly  occurring. 

Well-known  examples  of  the  unsettled  state  of  a  system  are: 
The  turbulent  state  with  its  different  streams,  whirls,  and  eddies, 
the  state  in  which  the  potential  and  kinetic  energy  is  unequally 
distributed;  for  instance,  when  one  part  is  at  a  high  pressure 
and  another  part  at  a  lower  pressure,  when  one  part  is  hotter 
than  another  part,  and  when  unmixed  gases  are  present  in  a 
communicating  system. 

A  more  specific  feature  of  the  unsettled  state  may  be  found 
in  the  accompaniment  to  BURBURY'S  condition  A  (already  men- 
tioned at  bottom  of  p.  15)  where  it  is  intimated  that  (at  the  start 
and  after  collision)  all  directions  of  the  relative  velocity  R  may 
not  be  equally  likely. 

When  such  differences  have  all  disappeared  to  the  extent  that 
equal  elementary  spaces  possess  their  equal  shares  of  the  different 
particles,  velocities,  and  energies,  the  system  will  be  a  settled 
one,  be  in  thermal  equilibrium,  and  will  possess  a  maximum 
entropy  and  a  definite  temperature.  Moreover,  BURBURY'S  con- 
dition A  is  here  fully  satisfied. 

At  this  point  we  again  call  attention  to  the  fact,  that  in  both 
the  unsettled  and  settled  states  of  a  system  all  conceivable  micro- 
states  are  not  equally  likely  to  obtain.  On  p.  9  mention 
was  made  that  the  unsettled  and  the  settled  state  each  pos- 
sessed a  host  of  conceivable  micro-states  which  agreed  with  the 
characteristic  averages  of  their  respective  macro-states  (the 
unsettled  and  the  settled  ones),  and  yet  in  each  set  some  of  these 
led  subsequently  to  events  which  did  not  accord  with  experience. 
Therefore  for  both  the  unsettled  and  the  settled  state  we  must 
limit  the  manifold  character  of  their  micro-states  by  eliminating 
all  those  micro-states  which  lead  to  results  contrary  to  experience. 
This  is  accomplished  by  assuming  the  hypothesis  of  "  elementary- 
disorder  "  (elementar-ungeordnet)  to  obtain  for  the  unsettled  as 
well  as  the  settled  state.  Now  so  far  as  the  haphazard  character 


AND  OF  THE  SECOND   LAW  21 

of  the  remaining  motions  are  concerned,  we  might  stop  right  here, 
for  the  very  nature  of  this  hypothesis  insures  results  in  harmony 
with  experience,  i.e.,  with  the  undisturbed  operation  of  the  laws 
of  probability. 

But  if  we  do  not  stop  here,  preferring  to  examine  some  of  the 
special  features  of  fortuitous  motion,  as  detailed  on  pp.  10, 13,  14 
and  17,  we  still  see  that  by  this  hypothesis  we  have  not  removed 
the  haphazard  character  of  the  remaining  motions  in  either  the 
unsettled  or  the  settled  state.  For  instance,  we  have  not 
removed  BURBURY'S  condition  A.  We  must  remember,  too,  that 
in  PLANCK'S  briefest  statement  of  "elementary  disorder"  (bot.  of 
p.  n),  two  important  features  of  haphazard  are  emphasized,  viz.: 
the  independence  and  great  number  of  the  constituents.  BOLTZ- 
MANN  in  his  Gas  Theorie  of  course  considers  the  special  features 
which  underlie  the  application  of  the  Calculus  of  Probabilities; 
thus  he  says  they  are,  the  great  number  of  molecules  and  the 
length  of  their  paths,  which  together  make  the  laws  of  the  colli- 
sion of  a  molecule  in  a  gas  independent  of  the  place  where  it 
collided  before.  Neither  has  the  introduction  of  the  hypothesis 
of  "elementary  disorder"  done  away  with  these  special  features. 
There  have  simply  been  excluded  trom  consideration  such  pre- 
computed  and  prearranged  regularities  in  the  paths  and  direc- 
tions of  molecules  as  purposely  interfere  with  the  operation  of 
the  laws  of  probability.  We  are  still  free  to  consider  all  the  imagin- 
able positions  and  velocities  of  the  individual  molecules  which  are 
compatible  with  the  mean  velocity,  density,  and  temperature 
properly  characteristic  of  each  stage  of  the  passage  from  the 
unsettled  to  the  settled  state.  For  adequate  haphazard  we  only 
need  the  assumption  that  the  molecules  fly  so  irregularly  as  to 
permit  the  operation  of  the  laws  of  probabilities.  Such  a  presen- 
tation as  this  of  course  calls  for  complete  trust  that  all  the  specified 
requirements  have  been  adequately  met  and  BOLTZMANN'S  emi- 
nence as  a  mathematical  physicist  and  the  endorsement  of  his  peers 
must  be  our  guarantee  for  such  confidence  and  trust. 


22  THE  PHYSICAL  SIGNIFICANCE  OF    ENTROPY 

Before  closing  this  discussion  of  unsettled  and  settled  states 
we  will  insert  here  two  remarks,  really  at  this  stage,  anticipatory  in 
their  nature.  The  first  is,  that  under  the  limitation  imposed  by  our 
supplementary  hypothesis  of  "  elementary  chaos,"  the  very  sharpest 
definition  of  any  macro-state  is  the  number  of  its  possible  micro- 
states.  This  is  evidently  the  number  of  permutations,  possible  with 
the  given  locus  and  velocity  elements  under  the  restriction  imposed 
above.  Later  on  we  will  find  that  this  number  of  possible  micro- 
states  is  smaller  for  the  unsettled  state  than  for  the  settled  one. 
This  gives  us  a  clean-cut  distinction  between  the  two  states  con- 
templated. The  second  remark  is  that  the  inevitable  change  in 
the  system  as  a  whole  is  always  from  the  less  probable  to  the  more 
probable,  is  a  passage  from  an  unsettled  state  of  the  system  to 
its  settled  state  and  this  is  here  synonymous  with  the  growth  of  the 
number  of  possible  micro-states.  It  is  this  difference  between  the 
initial  and  final  states  which  constitutes  the  universal  driving 
motive  in  all  natural  events. 


SECTION  B 

THE  APPLICATION  OF  CALCULUS  OF  PROBABILITIES 
IN  MOLECULAR  PHYSICS. 

(i)  The  Probability  Concept,  its  Usefulness  in  the  Past,  its  Present 
Necessity,  and  its  Universality. 

An  indication  of  its  essential  value  in  this  physical  discussion 
is  evidenced  by  the  fact  that  we  have  almost  unwittingly  been  forced 
to  constantly  refer  to  it  in  all  of  our  preliminaries.  But  when 
this  concept  is  first  broached  to  a  student,  he  feels  about  it  like 
the  "man  in  the  street";  it  is  by  the  latter  regarded  as  a  matter 
of  chance  and  hence  of  uncertainty  and  unreliability;  moreover, 
the  latter  knows  in  a  vague  way  that  the  subject  has  to  do  with 
averages,  that  it  is  often  of  a  statistical  nature,  and  knows  that 
statistics  in  general  are  widely  distrusted.  The  student  is  at 


AND   OF   THE  SECOND  LAW  23 

first  likely  to  share  these  views  with  said  man  in  the  street,  and 
at  best  feels  that  its  introduction  is  of  remote  interest,  far  fetched, 
and  tends  to  hide  and  dissipate  the  kernel  of  the  matter.  The 
student  must  disabuse  himself  of  these  false  notions  by  reflecting 
how  much  there  is  in  Nature  that  is  spontaneous,  in  other  words, 
how  many  events  there  are  in  which  there  is  a  passage  from  a 
less  probable  to  a  more  probable  condition  and  that  he  cannot 
afford  to  despise  or  ignore  a  Calculus  which  measures  these 
changes  as  exactly  as  possible. 

In  this  connection  BOLTZMANN  says:  (W.  S.  B.  d.  Akad.  d. 
Wiss.,  Vol.  LXVI,  B  1872,  p.  275). 

"The  mechanical  theory  of  heat  assumes  that  the  molecules 
of  gases  are  in  no  way  at  rest  but  possess  the  liveliest  sort  of  motion, 
therefore,  even  when  a  body  does  not  change  its  state,  every  one 
of  its  molecules  is  constantly  altering  its  condition  of  motion  and 
the  different  molecules  likewise  simultaneously  exist  side  by  side 
in  most  different  conditions.  It  is  solely  due  to  the  fact  that  we 
always  get  the  same  average  values,  even  when  the  most  irregular 
occurences  take  place  under  the  same  circumstances,  that  we  can 
explain  why  we  recognize  perfectly  definite  laws  in  warm  bodies. 
For  the  molecules  of  the  body  are  so  numerous  and  their  motions 
so  swift  that  indeed  we  do  not  perceive  aught  but  these  average 
values.  We  might  compare  the  regularity  of  these  average  values 
with  those  furnished  by  general  statistics  which,  to  be  sure,  are 
likewise  derived  from  occurrences  which  are  also  conditioned  by 
the  wholly  incalculable  co-operation  of  the  most  manifold  external 
circumstances.  The  molecules  are  as  it  were  like  so  many  indi- 
viduals having  the  most  different  kinds  of  motion,  and  it  is  only 
because  the  number  of  those  which  on  the  average  possess  the 
same  sort  of  motion  is  a  constant  one  that  the  properties  of  the 
gas  remain  unchanged.  The  determination  of  the  average  values 
is  the  task  of  the  Calculus  of  Probabilities.  The  problems  of 
the  mechanical  theory  of  heat  are  therefore  problems  in  this 
calculus.  It  would,  however,  be  a  mistake  to  think  any  uncertainty 


24  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

is  attached  to  the  theory  of  heat  because  the  theorems  of  probability 
are  applied.  One  must  not  confuse  an  imperfectly  proved  proposi- 
tion (whose  truth  is  consequently  doubtful)  with  a  completely  estab- 
lished theorem  of  the  Calculus  of  Probabilities;  the  latter  represents, 
like  the  result  of  every  other  calculus,  a  necessary  consequence 
of  certain  premises,  and  if  these  are  correct  the  result  is  confirmed 
by  experience,  provided  a  sufficient  number  of  cases  has  been 
observed,  which  will  always  be  the  case  with  Heat  because  of  the 
enormous  number  of  molecules  in  a  body." 

To  become  more  specific  we  will  mention  some  of  the  problems 
to  which  the  Theory  of  Probabilities  has  been  profitably  applied. 
In  business  to  life  and  fire  insurance;  in  engineering  to  reducing 
the  inevitable  errors  of  observations  by  the  Method  of  Least 
Squares;  and  in  physics  to  the  determination  of  Maxwell's  Law 
of  the  distribution  of  velocities.  The  results  thus  obtained  are 
universally  trusted  and  accepted  by  experts.  Why  then  should 
this  Calculus  not  be  applicable  to  the  more  general  natural  events  ? 

In  this  connection  consider  some  of  its  good  points:  (a)  It 
eliminates  from  a  problem  the  accidental  elements  if  the  latter 
are  sufficiently  numerous;  (b)  it  deals  legitimately  with  averages; 

(c)  it  involves  combination  considerations  other  than  averages; 

(d)  it  is  available  for  non-mechanical  as  well  as  mechanical  occur- 
rences and  thus  (e)  has  a  capacity  for  covering  the  whole  range 
of  natural  events^  giving  it  a  character  of  universality  which  is 
now  its  most  valuable  asset. 

As  an  example  of  this  we  may  instance  BOLTZMANN'S  deservedly 
famous  H-theorem,  which  establishes  the  one-sidedness  of  all 
natural  events.1  Concerning  it,  this  master  in  mathematical 
physics  says: 

"  It  can  only  be  deduced  from  the  laws  of  probability  that,  if  the 
initial  state  is  not  especially  arranged  ior  a  certain  purpose,  the 


1  The  H-theorem  considers  a  process  (consisting  of  a  number  of  separate, 
reversible  processes)  which  is  irreversible  in  the  aggregate. 


AND   OF  THE  SECOND  LAW  25 

probability  that  H  decreases  is  always  greater  than  that  it  increases. 
In  this  connection  we  may  add  that  BOLTZMANN  looked  forward 
to  a  time,"  when  the  fundamental  equations  for  the  motion  of 
individual  molecules  will  prove  to  be  merely  approximate  formulas, 
which  give  average  values  which,  according  to  the  Theory  of 
Probabilities,  result  from  the  co-operation  of  very  many  inde- 
pendently moving  individuals  constituting  the  surrounding 
medium,  for  example,  in  meteorology  the  laws  will  refer  only  to 
average  values  deduced  by  the  Theory  of  Probabilities  from  a 
long  series  of  observations.  These  individuals  must  of  course  be 
so  numerous  and  act  so  promptly  that  the  correct  average  values 
will  obtain  in  millionths  of  a  second." 

To  further  strengthen  our  faith  we  may  point  out  that  the 
probability  method  has  been  successfully  used  to  determine 
unique  results  from  complicated  conditions  and  has  been  employed 
for  the  general  treatment  of  problems.  In  the  case  before  us 
it  has  solved  the  entropy  puzzle  which  has  exercised  physicists, 
as  well  as  engineers,  for  decades,  and  it  has  thereby  emancipated 
the  Second  Law  from  all  anthropomorphism,  from  all  dependence 
on  human  experimental  skill.  When  we  take  the  broadest 
possible  view  of  its  character,  this  Calculus  enables  us  to  read 
the  present  riddle  of  our  universe,  namely,  why  it  is  in  its  present 
improbable  state.  We  have  therefore  in  this  Calculus  an  engine 
for  investigation  which  is  of  great  power  and  is  likely  to  play  a 
large  part  in  the  future  in  the  ascertainment  -of  physical  truth. 
Of  course  it  must  then  be  in  the  hands  of  masters.  It  is  they 
and  they  alone  who  can  properly  and  adequately  interpret  such 
a  physical  problem  as  the  one  before  us.  In  scientific  work  our 
last  court  of  appeal  must  be  Nature,  and  we  therefore  say:  The 
best  justification  for  the  use  of  the  Theory  of  Probabilities  in  our 
problem  is  that  its  results  are  in  such  complete  accord  with  the 
facts. 

In  dealing  with  this  physical  engine  of  investigation,  we  must 
again  call  attention  to  some  of  the  features  of  haphazard  necessary 


26  THE  PHYSICAL   SIGNIFICANCE   OF  ENTROPY 

for  its  legitimate  application.  Of  course  the  statement  of  these 
features  will  vary  with  the  mechanical  or  non-mechanical  character 
of  the  problem  to  which  it  is  applied.  As  we  are  here  dealing 
mainly  with  the  former,  we  will  limit  ourselves  to  its  features :  (a) 
The  elements  dealt  with  must  be  very  numerous,  strictly  speaking, 
infinite;  (b)  as  a  phase  of  (a)  we  may  say  also  that  when  we 
speak  of  the  probability  of  a  state  we  express  the  thought  that 
it  can  be  realized  in  many  different  ways;  (c)  when  we  speak 
of  the  relative  directions  of  a  pair  of  molecules  all  possible  direc- 
tions must  be  considered;  (d)  we  must  so  weight  the  elements 
say,  in  (a),  (&),  and  (c)  that  they  are  equally  likely;  (e)  every  one 
of  the  entering  elements  must  possess  constituents  of  which  each 
individual  is  independent  of  every  other;  for  instance,  (/)  in  a  gas 
the  place  where  a  molecule  collided  must  be  independent  of  the 
place  where  it  collided  before.  In  our  physical  problem  all  of 
these  features  are  not  always  realized;  for  instance,  the  number 
of  particles  of  gas  are  only  finite  instead  of  being  infinite; 
again,  all  relative  velocities  after  collision  of  a  pair  of  molecules 
are  not  equally  likely;  BOLTZMANN  and  BURBURY  provide  for  these 
shortcomings  by  very  truly  asserting  that  in  actual  cases  we  are 
not  dealing  with  isolated  systems,  that  the  surrounding  walls 
are  not  impervious  to  external  influences,  and  that  the  latter 
come  at  haphazard  without  regard  to  internal  state  of  the  system 
at  the  time,  thus  renewing  and  maintaining  the  desired  state  of 
haphazard. 

Methods.  This  Calculus  works  largely  by  the  determination 
of  averages  and  its  results  must  be  interpreted  accordingly. 
Moreover,  for  the  present  we  will  take  a  popular,  practical  view 
of  these  results  and  consider  a  very  great  improbability  as  equiva- 
lent to  an  impossibility.  Numerical  computations  are  essential 
in  most  uses  of  this  Calculus,  but  here  they  will  be  entirely  omitted. 


AND    OF    THE  SECOND    LAW  27 

(2)  What  is  Meant  by  the  Probability  of  a  State  ?    Example 

To  come  back  to  the  matter  in  hand  we  will  now  show  what 
is  here  meant  by  the  probability  of  any  state. 

When  we  speak  of  the  probability  W  of  a  particular  "  elementar- 
ungeordnete  "  state,  we  thereby  imply  that  this  state  may  be 
variously  realized.  For  every  state  (which  contains  many  like 
independent  constituents)  corresponds  to  a  certain  "  distribu- 
tion," namely,  a  distribution  among  the  gas  molecules  of  the 
location  co-ordinates  and  of  the  velocity  components.  But  such 
a  distribution  is  a  permutation  problem,  is  always  an  assignment 
of  one  set  of  like  elements  (co-ordinates,  velocity  components) 
to  a  different  set  of  like  elements  (molecules).  So  long  as  only  a 
particular  state  is  kept  in  view,  it  is  of  consequence  as  to  how 
many  elements  of  the  two  sets  are  thus  interchangeably  assigned 
to  each  other  and  not  at  all  as  to  which  individual  elements  of 
the  one  set  are  assigned  to  particular  individual  elements  of  the 
other  set.1  Then  a  particular  state  may  be  realized  by  a  great 
number  of  assignments  individually  differing  from  one  another, 
but  all  equally  likely  to  occur.2  If  with  PLANCK  we  call  such  an 
assignment  a  "  complexion,"  3  we  may  now  say  that  in  general  a 
particular  state  contains  a  large  number  of  different,  but  equally 
likely,  complexions.  This  number,  i.e.,  the  number  of  the  com- 
plexions included  in  a  given  state  can  now  be  defined  as  the  proba- 
bility W  of  the  state* 

Let  us  present  the  matter  in  still  another  form.  BOLTZMANN 
derives  the  expression  for  magnitude  of  the  probability  by  at 

1  For  an  example  of  such  permutations  see  pp.  28  and  61,  62. 

2  LIOUVILLE'S  theorem  is  the  criterion  for  the  equal  possibility  or  equal  proba- 
bility of  different  state  distributions. 

3  A  happy  term,  but  one  not  in  vogue  among  English-speaking  physicists. 

4  The  identity  of  entropy  with  the  logarithm  of  this  state  of  probability  W 
is  established  by  showing  that  both  are  equal  to  the  same  expression.     It  seems 
an  easy  step  from  this  derivation  to  BOLTZMANN'S  definition  of  entropy  as  the 
"measure  of  the  disorder  of  the  motions  in  a  system  of  mass  points." 


28  THE  PHYSICAL   SIGNIFICANCE   OF  ENTROPY 

once  distinguishing  between  a  state  of  a  considered  system  and 
the  complexion  of  the  considered  system.  A  state  of  the  system  is 
determined  by  the  law  of  locus  and  velocity  distribution,  i.e.,  by 
a  statement  of  the  number  of  particles  which  lie  in  each  ele- 
mentary district  of  space  and  the  number  of  particles  which  lie  in 
each  elementary  velocity  realm,  assuming  that  among  themselves 
these  districts  and  realms  are  alike  and  each  such  infinitesimal 
element  still  harbors  very  many  particles.  Accordingly  a  particu- 
lar state  of  the  system  embraces  a  very  large  number  of  com- 
plexions. For  if  any  two  particles  belonging  to  different  regions 
swap  their  co-ordinates  and  velocities,  we  get  thereby  a  new 
complexion,  but  still  the  same  state.  Now  BOLTZMANN  assumes  all 
complexions  to  be  equally  probable  and  therefore  the  number  of 
complexions  included  hi  a  particular  state  furnishes  at  the  same 
time  the  numerical  value  for  the  Probability  of  the  state  in 
question.  Illustration  taken  from  the  simultaneous  throwing  of 
two,  ordinary,  cubical  dice.  Suppose  that  the  sum  is  to  be  4  for 
each  throw,  then  this  can  be  realized  by  the  following  three  com- 
plexions: 

First  cube  shows  i,  the  second  cube  shows  3; 
First  cube  shows  2,  the  second  cube  shows  2; 
First  cube  shows  3,  the  second  cube  shows  i. 

The  requirement  that  the  sum  on  the  two  cubes  shall  be  2,  how- 
ever, involves  but  one  complexion.  Under  the  circumstances 
therefore  the  probability  of  throwing  the  sum  4  is  three  times 
as  great  as  throwing  the  sum  2. 

In  closing  this  part  of  our  presentation,  we  may  make  what  is 
now  an  almost  obvious  remark.  The  long-lasting  difficulty  in 
giving  a  physical  meaning  to  entropy  and  the  Second  Law  is  due 
to  the  fact  of  its  intimate  dependence  on  considerations  of  prob- 
ability. It  is  only  quite  recently  that  such  considerations  have 
attained  the  dignity  of  a  great  working  principle  in  the  domain  of 
Physics. 


AND   OF  THE  SECOND  LAW  29 


SECTION  C 

(i)  Existence,   Definition,   Measure,    Relations,    Properties,    and 
Scope  of  Irreversibitity  and  Reversibility. 

In  establishing  the  existence  of  irreversibility,  we  can  use 
one  or  both  of  the  two  general  methods  of  approaching  any  physical 
problems  (see  Introduction,  pp.  2,  3)  we  can  approach  by  way 
of  the  atomic  theory  or  by  considering  the  behavior  of  aggregates 
in  Nature.  Enough  has  already  been  said  in  this  presentation 
of  atomic  behavior  and  arrangements  to  justify  the  statement 
that  irreversibility  is  not  inherent  in  the  elementary  procedures 
themselves  but  in  their  irregular  arrangement.  The  motion 
of  each  atom  is  by  itself  reversible,  but  their  combined  mean 
effect  is  to  produce  something  irreversible.1 

This  has  been  rigorously  demonstrated  by  BOLTZMANN'S  H- 
theorem  for  molecular  physics,  and  when  sufficiently  general 
co-ordinates  are  substituted  it  is  also  available  for  the  other  domains 
of  natural  events.  When  we  consider  the  behavior  of  aggregates 
we  recognize  at  once  a  general,  empirical  law,  which  has  also 
been  called  the  one  physical  axiom,  namely,  that  all  natural 
processes  are  essentially  irreversible.  When  we  use  this  method 
of  approach  we  confessedly  rest  entirely  on  experience,  and  then 
it  does  not  make  any  logical  difference  whether  we  start  with 
one  particular  fact  or  another,  whether  we  start  with  a  fact  itself 
or  its  necessary  consequence:  For  instance  we  may  recognize 
that  the  universe  is  permanently  different  after  a  frictional  event 
from  what  it  was  before,  or  we  may  start,  as  PLANCK  does,  by 
putting  forward  the  following  proposition: 

"  //  is  impossible  to  construct  an  engine  which  will  work  in 


1  This  would  seem  to  imply  the  existence  of  a  broader  principle,  the  properties 
of  systems  as  a  whole  are  not  necessarily  found  in  their  parts. 


30  THE  PHYSICAL   SIGNIFICANCE  OF  ENTROPY 

a  complete  cycle,1  and  produce  no  effect  except  the  raising  of  a 
and  the  cooling  of  a  heat  reservoir"2 

Now  up  to  this  time  no  natural  event  has  contradicted  this 
theorem  or  its  corollaries.  The  proof  for  it  is  cumulative,  wholly 
experiential  and  therefore  exactly  like  that  for  the  law  of  con- 
servation of  energy. 

Returning  to  irreversibility,  the  matter  for  immediate  dis- 
cussion, we  premise  that  it  will  here  clarify  and  simplify  our 
ideas  if  we  consider  all  the  participating  bodies  as  parts  of  the 
system  experiencing  the  contemplated  process.  It  is  in  this 
v  sense  that  we  must  understand  the  statement :  Every  natural 
event  leaves  the  universe  different  from  what  it  was  before. 
Speaking  very  generally,  we  may  say  that  in  this  difference  lies 
what  we  call  irreversibility. 

Now  irreversibility  is  what  really  does  exist,  everywhere  in 
Nature,  and  our  idea  of  reversibility  is  only  a  very  convenient 
and  fruitful  fiction;  our  conception  of  reversibility  must,  there- 
fore, ultimately  be  derived  from  that  of  irreversibility. 

"  A  process  which  can  in  no  way  be  completely  reversed 
is  termed  irreversible,  all  other  processes  reversible.  That  a 
process  may  be  irreversible,  it  is  not  sufficient  that  it  cannot  be 
directly  reversed.  This  is  the  case  with  many  mechanical  processes 
which  are  not  irreversible  (See  p.  32).  The  full  requirement 
is,  that  it  be  impossible,  even  with  the  assistance  of  all  agents 
in  Nature,  to  restore  everywhere  the  exact  initial  state  when  the 

H        process  has  once  taken  place."/ 
,     Examples  of  irreversible  processes,  which  involve  only  heat 
and  mechanical  phenomena,  may  be  grouped  in  four  classes: 

(a)  The  body  whose  changes  of  state  are  considered  is  in 
contact  with  bodies  whose  temperature  differs  by  a  finite  amount 

1  Such  an  engine  if  it  would  work  might  be  called  "  perpetual  motion  of  the 
second  kind." 

2  The  term  perpetual  is  justified  because  such  an  engine  would  possess  the  most 
esteemed  feature  of  perpetual  motion — power  production  free  of  cost. 


AND   OF   THE  SECOND   LAW  31 

from  its  own.    There  is  here  flow  of  heat  from  the  hotter  to  the 
colder  body  and  the  process  is  an  irreversible  one. 

(b)  The    body    experiences    resistance    from    friction    which 
develops  heat;  it  is  not  possible  to  effect  completely  the  opposite 
operation  of  restoring  the  whole  system  to  its  initial  state. 

(c)  The  body  expands  without  at  the  same  time  developing 
an  amount  of  external  energy  which  is  exactly  equal  to  the  work 
of  its  own  elastic  forces.     For  example,  this  occurs  when  the 
pressure  which  a  body  has  to  overcome  is  essentially  (i.e.,  finitely) 
less  than  the  body's  own  internal  tension.     In  such  a  case  it  is 
not  possible  to  bring  the  whole  system  (of  which  the  body  is  a 
part)   completely  back  into  its  initial  state.     Illustrations  are: 
steam  escaping  from  a  high-pressure  boiler,  compressed  air  flowing 
into  a  vacuum  tank,  and  a  spring  suddenly  released  from  its 
state  of  high  tension. 

(d)  Two  gases  at  the   same  pressure  and  temperature  are 
separated  by  a  partition.     When  this  is  suddenly  removed,  the 
two  gases  mix  or  diffuse.    This  too  is  an  essentially  irreversible 
process. 

Outside  of  chemical  phenomena,  we  may  instance  still  other 
examples  of  irreversible  processes:  flow  of  electricity  in  conductors 
of  finite  resistance,  emission  of  heat  and  light  radiation,  and 
decomposition  of  the  atoms  of  radio-active  substances. 

"  Numerous  reversible  processes  can  at  least  be  imagined  y 
as,  for  instance,  those  consisting  throughout  of  a  succession  of 
states  of  equilibrium,  and  therefore  directly  reversible  in  all  their 
parts.  Further,  all  perfectly  periodic  processes,  e.g.,  an  ideal 
pendulum  or  planetary  motion,  are  reversible,  for,  at  the  end  of 
every  period  the  initial  state  is  completely  restored.  Also,  all 
mechanical  processes  with  absolutely  rigid  bodies  and  incom- 
pressible liquids,  as  far  as  friction  can  be  avoided,  are  reversible. 
By  the  introduction  of  suitable  machines  with  absolutely  unyield- 
ing connecting-rods,  frictionless  joints,  and  bearings,  inextensible 
belts,  etc.,  it  is  always  possible  to  work  the  machine  in  such  a 


32  THE   PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

way  as  to  bring  the  system  completely  into  its  initial  state  without 
leaving  any  change  in  or  out  of  the  machines,  for  the  machines 
of  themselves  do  not  perform  any  work." 

Other  examples  of  such  reversible  processes  are:  Free  fall  in 
a  vacuum,  propagation  of  light  and  sound  waves  without  absorp- 
tion and  reflection  and  unchecked  electrical  oscillations.  All  the 
latter  processes  are  either  naturally  periodic,  or  they  can  be  made 
completely  reversible  by  suitable  devices  so  that  no  sort  of  change 
in  Nature  remains  behind;  for  example,  the  free  fall  of  a  body 
by  utilizing  the  velocity  acquired  to  bring  the  body  back  to  its 
original  height,  light  and  sound  waves  by  suitably  reflecting  them 
from  perfect  mirrors. 

(2)  Character  of  Process  Decided  by  the  Limiting  States 

"  Since  the  decision  as  to  whether  a  particular  process  is  irre- 
versible or  reversible  depends  only  on  whether  the  process  can 
in  any  manner  whatsoever  be  completely  reversed  or  not,  the 
nature  of  the  initial  and  final  states,  and  not  the  intermediate 
steps  of  the  process,  entirely  settle  it.  The  question  is,  whether 
or  not  it  is  possible,  starting  from  the  final  state,  to  reach  the 
initial  one  in  any  way  without  any  other  change.  .  .  .  The  final 
state  of  an  irreversible  process  is  evidently  in  some  way  discrimi- 
nate from  the  initial  state,  while  in  reversible  processes  the  two 
states  are  in  certain  respects  equivalent.  .  .  .  To  discriminate 
between  the  two  states  they  must  be  fully  characterized.  Besides 
the  chemical  constitution  of  the  systems  in  question,  the  physical 
conditions,  viz.,  the  state  of  aggregation,  temperature,  and  pressure 
in  both  states,  must  be  known,  as  is  necessary  for  the  application 
of  the  First  Law." 

11  Let  us  consider  any  process  whatsoever  occurring  in  Nature. 
This  conducts  all  participating  bodies  from  a  particular  initial 
condition  A  to  a  certain  final  condition  B.  The  process  is 
either  reversible  or  irreversible,  any  third  possibility  being 


AND  OF   THE   SECOND  LAV/  33 

excluded.  But  whether  it  is  reversible  or  irreversible  depends 
solely  and  only  on  the  constitution  of  the  two  states  A  and  B, 
not  upon  the  other  features  of  the  course;  after  state  B  has  been 
attained,  we  must  here  simply  answer  the  question  whether  the 
complete  return  to  A  can  or  cannot  be  effected  in  any  manner 
whatsoever.  Now  if  such  complete  return  from  B  to  A  is  not 
possible  then  evidently  state  B  in  Nature  is  somehow  distin- 
guished from  state  A.  Nature  may  be  said  to  prefer  state  B  to 
state  A.  Reversible  processes  are  a  limiting  case;  here  Nature 
manifests  no  preference  and  the  passage  from  the  one  to  other 
can  take  place  at  pleasure,  in  either  direction.  [In  the  common 
case  of  isentropic  expansion  from  A  to  J5,  there  is  no  exchange 
of  heat  with  the  outside;  external  work  is  performed  at  the 
expense  of  the  inner  energy  of  the  expanding  body.  When 
state  B  is  attained  we  can  effect  a  complete  return  to  A  by  com- 
pressing isentropically,  thus  consuming  the  external  work  per- 
formed on  the  trip  from  A  to  B  and  restoring  the  internal  energy 
of  the  body.] 

"Now  it  becomes  a  question  of  finding  a  physical  magnitude 
whose  amount  will  serve  as  a  general  measure  of  Nature's  preference 
for  a  state.  This  must  be  a  magnitude  which  is  directly  determined 
by  the  state  of  the  contemplated  system,  without  knowing  any- 
thing of  the  past  history  of  the  system,  just  as  is  the  case  when 
we  deal  with  the  state's  energy,  volume,  etc.  This  magnitude 
would  possess  the  property  of  growing  in  all  irreversible  processes, 
while  in  all  reversible  processes  it  would  remain  unchanged. 
The  amount  of  its  change  in  a  process  would  furnish  a  general 
measure  for  the  irreversibility  of  the  process." 

"  Now  R.  CLAUSIUS  really  found  such  a  magnitude  and  called 
it  entropy.  Every  bodily  system  possesses  in  every  state  a 
particular  entropy,  and  this  entropy  designates  the  preference 
of  nature  for  the  state  in  question;  in  all  the  processes  which 
occur  in  the  system,  entropy  can  only  grow,  never  dimmish. 
If  we  wish  to  consider  a  process  in  which  said  system  is  subject  to 


34  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

influences  from  without,  we  must  regard  the  bodies  exerting 
such  influences  as  incorporated  with  the  original  system  and 
then  the  statement  will  hold  in  the  above  given  form." 

From  what  has  gone  before  it  is  evident  that  the  following 
commonly  drawn  conclusions  are  correct: 

An  irreversible  process  is  a  passage  from  a  less  probable  to 
a  more  probable  state  of  the  system. 

An  irreversible  process  is  a  passage  from  a  less  stable  to  a 
more  stable  state  of  the  system. 

An  irreversible  process  is  essentially  a  spontaneous  one,  inas- 
much as  once  started  it  will  proceed  without  the  help  of  any  external 
agency. 

We  have  in  a  general  way  reached  the  conclusion  that  entropy 
is  both  the  criterion  and  the  measure  of  irreversibility.  But  now 
let  us  become  more  specific  and  go  more  into  certain  details, 
namely,  the  common  features  in  all  irreversibility.  The  property 
of  irreversibility  is  not  inherent  in  the  elementary  occurrences 
themselves,  but  only  in  their  irregular  arrangement.  Irreversi- 
bility depends  only  on  the  statistical  property  of  a  system  possess- 
ing many  degrees  of  freedom,  and  is  therefore  essentially  based 
on  mean  values;  in  this  connection  we  may  repeat  an  earlier 
statement,  the  individual  motions  of  atoms  are  in  themselves 
reversible,  but  their  result  in  the  aggregate  is  not. 

(3)  All  the  Irreversible  Processes  Stand  or  Fall  Together 

This  is  proved  with  the  help  of  the  theorem  (p.  30)  which  denies 
the  possibility  of  perpetual  motion  of  the  second  kind.1  The 
argument  is  this:  take  any  case  in  any  one  of  the  four  classes  of 
irreversible  processes  given  on  p.  31.  Now  if  this  selected  case 

1  At  this  stage  we  appreciate  that  any  irreversible  process  is  a  passage  from  a 
state  A  of  low  entropy  to  a  state  B  of  high  entropy.  We  may  simplify  our  proof 
by  considering  the  return  passage  from  B  to  A  to  in  part  occur  isothermal ly 
and  in  part  isentropically;  then  external  agencies  must  produce  work  and  absorb 
an  equivalent  amount  of  heat. 


AND   OF   THE   SECOND   LAW  35 

is  in  reality  reversible,  i.e.,  suppose  a  method  were  discovered 
of  completely  reversing  this  process  and  thus  leave  no  other  change 
whatsoever,  then  combining  the  direct  course  of  the  process  with 
this  latter  reversed  process,  they  would  together  constitute  a 
cyclical  process,  which  would  effect  nothing  but  the  production 
of  work  and  the  absorption  of  an  equivalent  amount  of  heat. 
But  this  would  be  perpetual  motion  of  the  second  kind,  which 
to  be  sure  is  denied  by  the  empirical  theorem  on  p.  30.  But  for 
the  sake  of  the  argument  we  may  just  now  waive  said  impossibility; 
then  we  would  have  an  engine  which,  co-operating  with  any 
second  (so-called),  irreversible  process,  would  completely  restore 
the  initial  state  of  the  whole  system  without  leaving  any  other 
change  whatsoever.  Then  under  our  definition  on  p.  30  this 
second  process  ceases  to  be  irreversible.  The  same  result  will 
obtain  for  any  third,  fourth,  etc.  So  that  the  above  proposition 
is  established.  "  All  the  irreversible  processes  stand  or  fall  together." 
If  any  one  of  them  is  reversible  all  are  reversible.1 

(4)  Convenience  of  the  Fiction,  the  Reversible  Processes 

A  reversible  process  we  have  declared  to  be  only  an  ideal  case, 
a  convenient  and  fruitful  fiction  which  we  can  imagine  by  elim- 
inating from  an  irreversible  process  one  or  more  of  its  inevitable 
accompaniments  like  friction  or  heat  conduction.  But  reversible 
(as  well  as  irreversible)  processes  have  common  features.  "They 
resemble  each  other  more  than  they  do  any  one  irreversible  process. 
This  is  evident  from  an  examination  of  the  differential  equations 
which  control  them;  the  differential  with  respect  to  time  is  always 
of  an  even  order,  because  the  essential  sign  of  time  can  be  reversed. 
Then  too  they  (in  whatever  domain  of  physics  they  may  lie) 
have  the  common  property  that  the  Principle  of  Least  Action 

1  With  the  help  of  the  preceding  footnote  this  argument  can  be  followed  through 
in  detail  for  each  of  the  cases  enumerated  on  p.  31;  only  the  complicated  case  of 
diffusion  presents  any  difficulty. 


36  THE  PHYSICAL  SIGNIFICANCE   OF   ENTROPY 

can  represent  all  of  them  completely  and  uniquely  determines 
the  sequence  of  their  events."  They  are  useful  for  theoretical 
demonstration  and  for  the  study  of  conditions  of  equilibrium. 

There  is  a  certain,  limited,  incomplete  sense  in  which  we  say 
that  we  can  change  from  one  state  of  equilibrium  to  another  in 
a  reversible  manner.  For  example,  we  can,  considering  only  the 
one  converting  (or  intermediate)  body,  effect  said  change  by  a 
successive  use  of  isentropic  and  isothermal  change.  But  this 
ignores  all  but  one  of  the  participating  bodies  and  this  is  not 
permissible  if  we  strictly  adhere  to  the  true  definition  of  complete 
reversible  action. 

We  must  remember  too  that  no  other  universal  measure  of 
irreversibility  exists  than  entropy.  "Dissipation"  of  energy  has 
been  put  forward  as  such  a  measure,  but  we  know  already  of 
two  irreversible  cases  where  there  is  no  change  of  energy,  namely, 
diffusion  and  expansion  of  a  gas  into  a  vacuum.  [Unavailable, 
distributed,  scattered  energy  are  terms  which  could  be  used  here, 
free  from  all  objection.] 

But  of  course,  the  full  equivalent  of  entropy  can  be  substituted 
as  a  universal  measure  of  irreversibility.  On  p.  2  7  we  have  pointed 
out  that  the  number  of  complexions  included  in  a  given  state  can 
be  defined  as  the  probability  W  of  the  state,  then  in  a  footnote, 
attention  is  called  to  the  identity  of  entropy  with  the  logarithm 
of  this  state  of  probability  =  logarithm  of  the  number  of  complexions 
of  the  state.  This  makes  entropy  a  function  of  the  number  of 
complexions,  so  that  one  may  in  this  sense  be  regarded  as  the 
equivalent  of  the  other.  We  may  now  properly  speak  of  the 
number  of  complexions  of  a  state  as  the  universal  measure  of 
its  irreversibility.  The  physical  meaning  of  irreversibility  becomes 
apparent  when  put  in  this  form.  The  greater  the  number  of  com- 
plexions included  in  a  state  the  more  disordered  is  its  elementary 
condition  and  the  more  difficult  (more  impossible,  so  to  speak) ,  is 
it  to  directly  so  influence  the  constituents  of  the  whole  that  they 
will  reverse  the  sequence  of  the  mean  values  the  aggregate  tends 


AND   OF   THE  SECOND   LAW  37 

of  itself  to  assume.  An  illustration  will  help  to  make  this  clear; 
the  irreversible  case  in  which  work  (i.e.,  friction)  is  converted 
into  heat.  "  For  example,  the  direct  reversal  of  a  frictional  process 
is  impossible  because  this  would  presuppose  the  existence  of  an 
elementary  order  among  adjacent,  mutually  interacting  molecules. 
For  then  it  must  predominantly  be  the  case  that  the  collisions  of 
each  pair  of  molecules  must  bear  a  certain  distinguishable  char- 
acter inasmuch  as  the  velocities  of  two  colliding  molecules  must 
always  depend  in  a  determinate  manner  on  the  place  where  they 
meet.  Only  thereby  can  it  be  attained  that  there  will  result 
from  the  collisions  predominantly  like  directed  velocities." 

The  outcome  of  the  whole  study  of  irreversibility  results  in 
the  briefly  stated  law  :  "  There  exists  in  Nature  a  quantity  which 
changes  always  in  the  same  sense  in  all  natural  processes" 

This  boldly  asserts  the  essential  one-sidedness  of  Nature. 
The  proposition  stated  in  this  general  form  may  be  correct  or 
incorrect;  but  whichever  it  may  be  it  will  remain  so  independently 
of  human  experimental  skill. 

SECTION  D 

(i)  The  Gradual  Development  of  the  Idea  that  Entropy  Depends 

on  Probability 

Entropy  is  difficult  to  conceive,  in  that,  as  it  does  not  directly 
affect  the  senses,  there  is  nothing  physical  to  represent  it;  it 
cannot  be  felt  like  temperature.  It  has  no  analogue  in  the  whole 
of  Physics;  Zeuner's  heat  weight  will  perhaps  serve  as  such  for 
reversible  states,  but  is  inadequate  for  irreversible  ones.  This 
is  not  surprising  when  we  consider  the  outcome,  namely,  that  it 
depends  on  probability  considerations. 

CLAUSIUS  coined  the  term  Entropy  from  the  Greek,  from  a 
word  meaning  transformation;  with  him  the  transformation  value 
was  equal  to  the  difference  between  the  entropy  of  the  final  and 
initial  states.  As  there  is  a  general  expression  for  entropy,  we 


38  THE  PHYSICAL  SIGNIFICANCE   OF  ENTROPY 

can  readily  write  the  equivalent  of  any  transformation  between 
two  particular  states. 

Strictly  speaking,  however,  entropy  by  itself  depends  only  on 
the  state  in  question,  not  on  any  change  it  may  experience,  nor 
on  its  past  history  before  reaching  the  state  contemplated.  Of 
course,  this  was  appreciated  by  such  a  master  mind  as  CLAUSIUS, 
and,  indeed,  he  defined  the  entropy  as  the  algebraic  sum  of  the 
transformations  necessary  to  bring  a  body  into  its  existing  state. 
Moreover,  as  the  formula  for  it  was  in  terms  of  other  more  or 
less  sensible  thermodynamic  quantities,  its  relation  to  these  was 
at  first  more  readily  grasped,  could  be  represented  diagrammat- 
ically,  and  had  to  do  duty  for  the  true,  but  still  unknown,  physical 
idea  of  entropy  itself.  It  was  early  understood,  too,  that  growth 
of  entropy  was  closely  connected  with  the  degradation  or  waste 
of  energy;  that  it  was  identical  with  the  Second  Law.  The  fre- 
quently given,  but  not  always  valid,  relation, 

dQ-TdS* 

led  to  entropy  being  called  a  factor  of  energy.  But  all  these 
were  change  relations  and  did  not  go  to  the  root  of  the  difficulty, 
as  to  what  constituted  the  physical  nature  of  unchanged  entropy. 
Quite  early,  too,  there  was  a  realization  of  the  fact  that 
entropy  had  somehow  a  statistical  character,  that  it  had  to  do 
with  mean  values  only.  This  was  well  brought  out  by  the  long 
known,  and  much  quoted,  "  demon  "  experiment  suggested  by 
Maxwell,  in  which  a  being  of  superhuman  power  separated, 
without  doing  any  work,  the  colder  and  hotter  particles  of  a  gas, 
thus  effecting  an  apparent  violation  of  the  Second  Law.  This,  to 
be  sure,  was  getting  close  to  the  crux  of  the  whole  matter,  but 
still  lacked  much  to  give  entropy  a  precise  physical  meaning. 
Nevertheless,  we  see  here  a  notable  approach  to  the  fundamental 

1  This  relation  is  not  a  valid  one,  unless  the  external  work  performed  by  a  gas 
during  its  change  is  equal  to  pdV. 


AND   OF   THE   SECOND  LAW  39 

requirement  that  entropy  must  be  tied  down  to  the  condition  of 
"  elementary  chaos  "  (elementare-unordnung). 

We  have  already  dwelt  somewhat  fully  on  this  hypothesis  of 
"  elementary  chaos." 

"  It  follows  from  this  presentation  that  the  concepts  of  entropy 
and  temperature  in  their  essence  are  tied  to  the  condition  of 
"  elementare  Unordnung."  Thus  a  purely  periodic  absolute 
plane  wave  possesses  neither  entropy  nor  temperature  because 
it  contains  nothing  whatever  in  the  way  of  uncheckable,  non- 
measurable  magnitudes,  and  therefore  cannot  be  "  elementar- 
ungeordnet,"  just  as  little  as  can  be  the  case  with  the  motion  of 
a  single  rigid  atom.  When  there  is  [an  irregular  co-operation  of 
many  partial  oscillations  of  different  periods,  which  independently 
of  each  other  propagate  themselves  in  the  different  directions  of 
space,  or]  an  irregular,  confused,  whirring  intermingling  of  many 
atoms,  then  (and  not  till  then)  is  there  furnished  the  preliminary 
condition  for  the  validity  of  the  hypothesis  of  "  elementare  Unord- 
nung and  consequently  for  the  existence  of  entropy  and  of 
temperature." 

"  Now  what  mechanical  or  electro-dynamic  magnitude  repre- 
sents the  entropy  of  a  state?  Evidently  this  magnitude  depends 
m  some  way  on  the  "  Probability  "  of  the  state.  For  because 
"  elementare  Unordnung  "  and  the  lack  of  every  individual  check 
(or  measurement)  is  of  the  essence  of  entropy  it  follows  that  only 
combination  or  probability  considerations  can  furnish  the  necessary 
foothold  for  the  computation  of  this  magnitude.  Even  the 
hypothesis  of  "  elementare  Unordnung  "  by  itself  is  essentially 
a  proposition  in  Probability,  for,  out  of  a  vast  number  of  equally 
possible  cases,  it  selects  a  definite  number  and  declares  they  do 
not  exist  in  Nature." 

Now  since  the  idea  of  entropy,  and  likewise  the  content  of 
Second  Law,  is  a  universal  one,  and  since,  moreover,  the 
theorems  of  probability  possess  no  less  universal  significance,  we 
may  conjecture  (surmise)  that  the  connection  between  Entropy 


-^  40  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

and  Probability  will  be  a  very  close  one.  We  therefore  place 
at  the  head  (forefront)  of  our  further  presentation  the  following 
proposition:  "  The  Entropy  of  a  physical  system  in  a  definite 
condition  depends  solely  on  the  probability  of  this  state"  The 
permissibility  and  fruitfulness  of  this  proposition  will  become 
manifest  later  in  different  cases.  A  general  and  rigorous  proof  of 
this  proposition  will  not  be  attempted  at  this  place.  Indeed, 
such  an  attempt  would  have  no  sense  here  because  without  a 
numerical  statement  of  the  probability  of  a  state  it  could  not  be 
tested  numerically. 
. 

(6)  Planck's  Formula  for  the  Relation  between  Entropy  and  the 
Number  of  Complexions 

Now  we  have  already  seen,  from  the  permutation  consider- 
ations presented  on  p.  27,  that  the  Theory  of  Probabilities  leads 
very  directly  to  the  theorem,  "  The  number  of  complexions  included 
in  a  given  state  constitutes  the  probability  W  of  that  state."  The 
next  step  (omitted  here)  is  to  identify  the  thermodynamically 
found  expression  for  entropy  of  any  state  with  the  logarithm 

of  its  number  of  complexions. 

-1 

PLANCK'S  formula  for  entropy  S  is: 

5  =  1.35  loge  (number  of  complexions)  io~16+ constant  K; 

here  K  is  an  arbitrary  constant  without  physical  significance 
and  can  be  omitted  at  pleasure;  the  numerical  value  in  the  first 
term  of  the  second  member  is  the  quotient  of  energy  (expressed 
in  ergs)  divided  by  temperature  (C^.  This  certainly  gives  a  phys- 
-•A*  ical  definiteness  and  precision  to  entropy  which  leaves  nothing 
to  be  desired. 

PLANCK,  in  reproducing  from  probability  consideration  the 
dependence  of  entropy  5  on  probability  W,  finds  the  relation 

5  =  &  log  W + constant, 
when  the  dimensions  of  5  evidently  depend  on  those  of  constant  k. 


AND   OF   THE  SECOND    LAW  41 

Here  S  is  BOLTZMANN'S  value  —  H,  wrn'ch  always  changes 
in  one  direction  only;  k  is  the  universal  integration  constant 
which  is  the  same  for  a  terrestrial  as  for  a  cosmical  system,  and 
when  it  is  known  for  one,  it  is  known  for  the  other;  when  k  is 
known  for  radiant  phenomena  it  is  also  known  and  is  the  same 
for  molecular  motions. 

There  are  some  general  statements  which  indicate  more  or 
less  rigorously  some  of  the  properties  or  features  of  the  entropy 
of  a  state. 

(a)  Entropy  is  a  universal  measure  of  the  "  disorder  "  in  the 

mass  points  of  a  system. 

(b)  Entropy  is  a  universal  measure  of  the  irreversibility  of  a 

state  and  is  its  criterion  as  well. 

(c)  Entropy  is  a  universal  measure  of  nature's  preference  for 

the  state. 

(d)  Entropy  is  a  universal  measure  of  the  spontaneity  with 

which  a  state  acts  when  it  is  free  to  change. 

(e)  Entropy  of  a  system  can  only  grow. 

(f)  Entropy  asserts  the  essential  one-sidedness  of  Nature. 

(g)  There  exists  in  Nature  a  magnitude  which  always  changes 

in  the  same  sense. 

(e),  (/),  and  (g)  imply  change  and  therefore,  strictly  speaking, 
should  not  be  mentioned  here  but  postponed  to  a  later  section. 


SECTION  E 

EQUIVALENTS   OF   CHANGE   OF    ENTROPY  IN   MORE   OR   LESS 
GENERAL   PHYSICAL  TERMS 

Here  we  are  really  considering  the  Second  Law,  for  change 
of  entropy  is  the  kernel  of  this  law,  in  fact  is  identical  with  it. 
It  will  be  profitable,  however,  to  view  this  law  in  all  its  many 
physical  aspects.  To  be  sure,  in  times  past  it  has  been  accounted 
a  reproach  to  the  Second  Law  that  it  should  be  stated  in  so  many 


42  THE   PHYSICAL   SIGNIFICANCE   OF  ENTROPY 

different  forms,1  but  now  that  we  know  precisely  that  it  stands 
for  the  growth  in  the  number  of  complexions  we  can  more  easily 
trace  the  connection  between  any  of  these  rather  vague  state- 
ments and  the  present  precise  definition.  As  we  have  in  the 
main  reserved  physical  interpretations  to  a  later  section  we 
need  here  only  bear  in  mind  certain  general  principles  of  com- 
parison : 

Any  complete  summary  of  the  premises  necessary  for  estab- 
lishing the  inevitable  growth  of  the  number  of  complexions  of 
a  system  is  a  valid  statement  of  the  second  law. 

Any  general  corollary  from  said  growth  is  a  valid  statement 
of  the  second  law. 

When  instituting  any  comparison  we  must  keep  in  mind  also 
the  two  principal  points  of  view  of  regarding  any  physical  problem, 
namely,  the  view  of  it  in  the  aggregate  and  that  which  sees  it  in 
its  constituent  parts. 

While  we  cannot  here  sharply  separate  these  two  points  of 
view,  we  have  on  the  whole  sought  to  present  first  those  statements 
which  are  based  on  experience  and  next  those  based  on  the  atomic 
theory. 

(1)  Growth  of  entropy  is  a  passage  from  more  to  less  avail- 
able energy.     By  available  is  here  meant  energy  which 
we  can  direct  into  any  required  channel.     With  the  growth 
in  the  number  of  complexions  we  can  readily  see  there 
is  greater  inability,  on  the  part  of  the  molecules,  for  that 
concerted  and  co-operative  action  which  is  necessary  for 
the  putting  forth  of  the  energy  of  a  system. 

(2)  Growth  of  entropy  is  a  passage  from  a  concentrated  to  a 
distributed  condition   of  energy.     Energy   originally  con- 
centrated variously  in  the  system  is  finally  scattered  uni- 

1  This  need  cause  no  surprise,  for  it  is  only  very  recently  that  the  conviction 
is  gaining  ground  that  the  Second  Law  has  no  independent  significance,  but  that 
its  full  content  will  only  he  grasped  when  its  roots  are  sought  in  the  Theorems  of 
the  Calculus  of  Probabilities. 


AND  OF   THE  SECOND   LAW  43 

formly  in  said  system.     In  this  aggregate  aspect  it  is  a 
passage  from  variety  to  uniformity. 

(3)  Net  growth  of  entropy  in  all  bodies  participating  in  an 
occurrence  means  that  the  system  as  a  whole  has  experienced 
an  irreversible  change  of  state.    This  change  is  of  course 
in  harmony  with  the  first  law  of  energy,  but  this  growth 
gives  additional   information  as  it  indicates  the  direction 
in  which  a  natural  process  occurs. 

(4)  Growth  of  entropy  is  from  less  probable  to  more  probable 
states. 

Growth  of  entropy  is  passage  to  a  state  more  greatly  preferred 

by  nature. 
Growth  of  entropy  is  what  obtains  whenever  a  natural  process 

occurs  "  spontaneously." 

f  unsettled   1 
Growth  of  entropy  is  a  passage  from  j ,  . .     Y  to  more 

f  settled  1 

\    j.  LI      f  conditions. 

[  stable  J 

All  these  statements  are  conspicuously  based  on  the  theory 
of  probabilities. 

(5)  Growth  of  entropy  is  a  passage  from  a  somewhat  regulated 
to  a  less  regulated  state.    It  represents,  in  a  certain  sense, 
Nature's  escape  from  thralldom. 

Growth  of  entropy  is  a  passage  from  a  somewhat  ordered 

to  a  less  ordered  molecular  arrangement. 
Growth  of  entropy  is  an  increase  in  the  disorder  of  a  system 

of  mass  points. 
Growth  of  entropy  corresponds  to  an  increase  in  the  number 

of  molecular  complexions. 

(6)  Finally  we  give  a  mathematical  concept  which    covers 
the  whole  domain  of  physics:  "  Any  function  whose  time 
variation  always  has  the  same  sign  until  a  certain  state 
is  reached  and  is  then  zero,  may  be  called  an  entropy 
function." 


44  THE  PHYSICAL  SIGNIFICANCE   OF   ENTROPY 

SECTION   F 
MORE  PRECISE  AND  SPECIFIC  STATEMENTS  OF  THE  SECOND  LAW 

We  have  here  classified  these  statements  in  the  same  way  as 
that  followed  in  the  preceding  section,  when  grouping  the  general 
equivalents  of  the  Second  Law  under  the  head  of  change  of  entropy. 
In  making  comparisons  we  must,  here  as  there,  bear  in  mind  the 
following  three  helpful  propositions : 

(a)  The  summary  of  all  the  necessary  prerequisites  (or  condi- 
tions) for  determining^  entropy  may  be  regarded  as  a  complete 
and  valid  statement  of  the  second  law. 

(b)  Any  general  consequence  of  any  one  correct  statement 
of  the  second  law  may  be  regarded  as  itself  a  valid  and  complete 
statement  of  the  second  law. 

(c)  All  cases  of  irreversibility  stand  or  fall  together;    if  any 
one  of  them  can  be  completely  reversed  all  can  be  so  reversed. 

In  the  preceding  section  we  have  already  given  the  most 
precise  physical  statement  of  the  Second  Law,  namely,  when  all 
the  participating  bodies  of  the  system  are  considered,  every 
natural  event  is  marked  by  an  increase  in  the  number  of  com- 
plexions of  the  system.  We  have  numbered  the  following  state- 
ments of  the  second  law,  for  convenience  of  reference: 

(1)  J.  W.  GIBBS.      "The  imposliibility  of  an  uncompensated 
decrease  in  entropy  seems  to  be  reduced  to  an  improbability." 
This   of  course   considers  all  the  participating  bodies  of  the 
system. 

(2)  All  changes  in  nature  involve  a  net  growth  in  entropy; 
when  such  a  change  is  measured  in  reversible  ways,  the  growth 

is  indicated  by  the  summation :     I  —  ^  o,  when  the   \ ,  [• 

sign    refers    to  processes  which    on  the  whole  are  completely 

{., ,      k     Of  course  it  is  now  thoroughly  understood  that 
reversible  ' 


AND    OF   THE  SECOND   LAW  45 

the  latter  case  is  a  purely  ideal  one,  which  is  really  never  realized 
in  nature  and  is  only  a  convenient  and  fruitful  fiction  in  theoretical 
demonstrations. 

(3)  M.  PLANCK.  "  It  is  not  possible  to  construct  a  periodically 
functioning  motor  which  effects  nothing  more  that  the  lifting 
of  a  load  and  the  cooling  of  a  heat  reservoir." 

The  proof  of  this  is  purely  experimental  and  cumulative 
and  in  this  respect  is  exactly  like  that  for  the  First  Law,  the  Con- 
servation of  Energy,  and  has  exactly  the  same  sort  of  validity. 

(4)  Perpetual  motion  of  an  isolated  system,  such  as  a  mechan- 
ism with  friction,  is  impossible  and  not  even  approximately  real- 
izable. 

This  refers  to  perpetual  motion  of  the  second  class,  a  clear 
illustration  of  which  is  given  on  p.  8  of  Goodenough's  Notes  on 
Thermodynamics:  "A  mechanism  with  friction  is  inclosed  in 
a  case  through  which  no  energy  passes.  Let  the  mechanism  be 
started  in  motion.  Because  of  friction  work  is  converted  into 
heat  which  remains  in  the  system,  since  no  energy  passes  through 
the  case.  Suppose  that  the  heat  thus  produced  could  be  completely 
transformed  into  work;  then  this  work  would  be  used  again  to 
overcome  friction  and  the  heat  thus  produced  would  be  again 
transformed  into  work.  We  should  then  have  perpetual  motion 
in  a  mechanism  with  friction  without  the  addition  of  energy 
from  an  external  source."  This  can  be  shown  to  be  equivalent 
but  not  identical  with  the  "  perpetual  motion  of  the  second  kind," 
touched  upon  in  p.  30;  the  latter  does  confessedly  draw  on 
external  energy  and  furnishes  a  surplus  of  power  for  use,  say,  in 
technical  service. 

Nominally,  such  a  machine  is  a  case  of  perpetual  motion,  but 
not  in  the  usually  accepted  sense,  for  it  furnishes  no  surplus  of 
power;  it  is  the  getting  of  something  for  nothing,  of  getting  cost- 
free  power,  which  has  always  been  the  attractive  feature  of  so-called 
perpetual  motion.  Still  this  machine  is  as  much  at  variance 
with  experience  as  PLANCK'S  perpetually  working  motor  of  the 


7 


46  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

second  kind.  The  former  may  be  readily  reduced  to  the  latter, 
for  it  is  easy  to  conceive  of  such  legitimate  modification  of  the 
former  as  will  make  it  only  a  special  case  of  the  latter. 

(5)  The  following  statements  are  by  distinguished  physicists 
and  had  better  here  be  considered  as  confined  to  events  occurring 
in  closed  cycles. 

CLAUSIUS.  It  is  impossible  for  a  self-acting  machine  unaided 
by  any  external  agency  to  convey  heat  from  one  body  to  another 
of  higher  temperature. 

CLERK  MAXWELL.  It  is  impossible  by  the  unaided  action  of 
$£  natural  processes  to  transform  any  part  of  the  heat  of  a  body 
into  mechanical  work,  except  by  allowing  heat  to  pass  from  that 
body  to  another  of  lower  temperature. 

THOMSON.  It  is  impossible  by  means  of  inanimate  material 
agency  to  derive  mechanical  effect  from  any  portion  of  matter 
by  cooling  it  below  the  temperature  of  the  coldest  of  surrounding 
objects. 

(6)  The  efficiency  of  a  perfect  engine  is  independent  of  the 
working  fluid. 

(7)  Waste  of  energy  once  incurred  cannot  be  diminished  in 
the  universe,  or  in  any  part  of  it  which  neither  takes  in  nor  gives 
out  energy. 

We  understand  here  by  waste  that  residual  part  of  heat  of 
which  none  can  be  elevated  back  into  work. 

The  measure  of  such  ^waste  =  To(S2  —  Si),  when  TO  =  lowest 
temperature  and  S2—Si  =  change  of  entropy  in  a  process.  This 
brings  out  emphatically  that  the  Second  Law  is  not  a  law  of  con- 
servation, it  is  a  law  of  waste,  a  law  of  wasted  opportunities  for 
utilizing  technically  available  energy. 

(8)  The  second  law  and  irreversibility  do  not  depend  on  any 
special  peculiarity  of  heat  motion,  but  only  on  the  statistical 
property  of  a  system  possessing  an  extraordinary  number  of 
degrees  of  freedom. 

(9)  M.  PLANCK.     The  second  law,  in  its  objective  physical 


AND  OF   THE   SECOND   LAW  47 

form  (freed  from  all  anthropomorphism)  refers  to  certain  mean 
values  which  are  formed  from  a  great  number  of  like  "  chaotic  " 
elements. 

(10)  When  all  the  participating  bodies  of  the  system  are 
considered,  every  natural  event  is  marked  by  an  increase  in  the 
number  of  complexions  of  the  system.  We  repeat,  this  is  the 
most  precise  physical  statement  of  the  second  law  and  covers  the 
whole  domain  of  science. 

We  will  not  comment  further  on  these  statements  at  this  time, 
leaving  such  discussion  of  their  relations  to  the  section  on  physical 
interpretations. 


48  THE  PHYSICAL   SIGNIFICANCE  OF  ENTROPY 


PART  II 

ANALYTICAL  EXPRESSIONS  FOR  A  FEW  PRIMARY  RELATIONS 

AT  the  beginning  of  this  presentation  we  disclaimed  any  pur- 
pose of  giving  a  rigorous  proof  for  any  of  the  many  formulas 
with  which  this  subject  bristles.  We  propose  only  to  give  in  some 
cases  an  outline  of  the  main  steps  of  the  demonstration  and 
merely  for  the  purpose  of  getting  a  clearer  physical  insight  into 
certain  states  and  relations.  Pre-eminent  in  importance  is  the 
state  of  thermal  equilibrium  (see  pp.  19,  52,  53)  and  we  will 
therefore  consider  first  its  main  characteristic: 

SECTION  A 
MAXWELL'S  LAW  OF  DISTRIBUTION  OF  MOLECULAR  VELOCITIES 

Without  giving  a  full  proof  of  the  law  we  will  give  the  main 
steps  which  lead  to  its  analytical  statement,  in  so  doing  following 
the  presentation  given  by  HANS  LORENZ  on  pp.  526-529  of  his 
"  Technische  Warmelehiie,"  and  will  then  point  out  its  main 
features  and  consequences. 

We  suppose  the  gas  to  contain  in  a  unit  of  volume  n  molecules 
each  possessing  a  different  velocity  and  direction.  Let  there  be 
a  system  of  three  co-ordinate  axes,  £,  77,  £.  A  fraction  f(£)d£ 
of  the  total  number  of  molecules  will  possess  a  velocity  in  the  £ 
direction,  whose  values  lie  between  £  and  £  +  d£.  The  number 
of  molecules  which  at  the  same  time  possess  velocities  in  the  T? 
direction,  lying  between  TJ  and  y+dy,  will  be  nf(£)d£f(r))dt), 
since  no  preference  can  be  given  to  either  the  £  or  r)  direction. 
Similarly  and  finally  the  number  of  molecules  whose  velocity 


AND   OF    THE  SECOND  LAW  49 

co-ordinates  concurrently  lie  between  £,  77,  £,  and  £+</£, 
will  be  represented  by  the  product 


where  the  only  thing  known  about  function  /  is  that  the  sum  of 
the  fractions  f(£)dg  extended  over  all  the  values  of  £  must  =  unity, 
so  that 


Now  if  we  suppose  all  of  the  velocities  of  the  n  molecules  to 
be  laid  off  as  vectors  from  a  pole  O,  the  three  directions  £,  9, 
£  will  constitute  about  O  a  perfectly  arbitrary  system  of  co-ordi- 
nates in  which  (d£)  (dy)  (dV)=dV  designates  a  volume  element  l 
and  the  velocity  p  of  a  molecule  is  given  by 


1  In  MAXWELL'S  distribution  the  molecules  are  assumed  to  be  uniformly  scat- 
terred  throughout  the  unit  volume;  it  is  the  velocities  only  that  are  variously 
distributed  in  the  different  elementary  regions.  To  realize  the  haphazard  char- 
acter (necessary  in  Calculus  of  Probabilities)  of  the  motions  of  the  molecules,  we 
must  bear  in  mind  that  each  of  the  molecules  in  the  unit  volume  has  a  different 
velocity  and  direction;  here  no  direction  has  preference  over  another,  i.e.,  one  direc- 
tion of  a  molecule  is  as  likely  as  another.  Here  at  first  we  write  expression  for 
the  number  of  molecules  whose  velocities  parallel  to  the  co-ordinate  axes  are 
respectively  confined  between  the  velocity  limits: 

£    and 
T)     and 

C    and 

To  find  the  number  of  molecules  thus  limited  the  procedure  given  above  isessentially 
as  follows:  Expressed  as  a,  fraction  f(£)d£=  probability  of  velocities  parallel  to  £ 
axis  having  values  between  £  and  £  +  d£  and  expressed  as  a  number  nf(£)d£= 
number  of  molecules  having  such  velocities  between  the  assigned  limits;  similarly, 
f(rj)dr)=  probability  of  velocities  parallel  to  TJ  axis  having  velocities  between  TJ  and 
ij+di).  As  these  are  two  independent  sets  of  velocities,  the  probability  of  their 
concurrence  is  the  product  f(£)d£-f(i))di)  and  the  number  of  molecules  thus  con- 


50  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

Now  if  we  put  through  the  origin  O  another  system  of  co-ordi- 
nates of  which  one  axis  coincides  with  any  arbitrarily  chosen 
velocity  p,  then  in  this  axis  the  above-found  product  will  be 
nf(p)f(O)  -f(O)dV  because  the  two  other  co-ordinates  (outside 
of  p)  of  the  volume  element  dV  will  equal  zero  and  no  preference 
can  be  given  to  any  direction.  Then  it  can  be  shown  that  the 
form  of  the  function  is  given  by 

_i! 

Ce   <V (3) 


where  C,  c  are  integration  constants  which  stand  in  a  certain 
relation  to  each  other,  namely, 

C  =  ~^.  (4) 


Further  mathematical  manipulation  eliminates  the  different 
velocity  directions  and  gives 


,  ...     (5) 


for  the  number  of  molecules  possessing  absolute  velocities  between 
p  and  p-\-dp. 

This  expression  (5)  is  called  MAXWELL'S  Law  of  Distribution; 
it  is  identical  with  that  found  for  the  probable  distribution  of 
'error  in  a  great  number  of  observations  and  is  graphically  shown 
by  the  following  figure,  with  maximum  number  of  molecules  for 
velocity  c.  The  constant  c  is  therefore  a  velocity  from  which 

curring  is  equal  to  nf(£)d£  -f(TJ)d£.     Similarly,  the  number  of  molecules  concurrently 
possessing  velocities  parallel  to  each  of  the  three  axes  is 


The  problem  is  simpler  in  this  Maxwellian  case  than  in  the  more  general  case  of 
any  state  of  the  body  in  which  there  is  an  unequal  distribution  in  space  of  the 
molecules. 


AND  OF  THE  SECOND  LAW 


51 


most  of  the  molecules  differ  but  little.  The  development  shows 
that  this  self-same  distribution  exists  for  every  straight  line  that 
can  be  drawn  in  the  volume  under  consideration. 


3  5 

if 


• 

E  = 


Velocities.  ' 

BOLTZMANN,  in  his  Gas  Theorie,  has  shown  that  for  such  a 
state  the  "  number  of  complexions  "  is  a  maximum,  that  is,  the 
entropy  is  then  a  maximum. 

From  the  preceeding  expression  (5)  follows  that  the  kinetic 
energy  U  of  the  system  is 


U 

where  [w2]  is  the  mean  square  of  the  velocity.     Integration  gives 

f*  (7) 


It  is  known  that  the  measure  of  temperature  is  the  mean  kinetic 
energy  of  the  individual  molecule  and  not  simply  the  mean  square 
of  its  velocity,  and  we  possess  here  therefore  a  perfectly  precise 


52  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

definition  of  temperature.  We  see  also  from  (7)  that  temperature 
in  a  particular  gas  is  directly  proportional  to  either  c2  or  [w2]. 

MAXWELL  further  shows  without  any  assumption  as  to  the 
nature  of  the  molecules,  or  the  forces  acting  between  them,  that 
the  derived  law  of  distribution  is  valid  for  any  gas  mixture,  but 
that  is  it  modified  when  the  gas  is  exposed  to  the  action  of  external 
forces. 

BOLTZMANN  found  (Wien.  Akad.  Sitzber.  LXXII  B,  1875,  p. 
443)  for  monatomic  gases  that  in  spite  of  the  effect  of  external 
forces  (a)  the  velocity  of  any  molecule  is  equally  likely  to  have 
any  direction  whatever,  (b)  the  velocity  distribution  in  any  element 
of  space  is  exactly  like  that  in  a  gas  of  equal  density  and  temperature 
upon  which  no  external  forces  act,  the  effect  of  the  external  forces 
consisting  only  in  varying  the  density  from  place  to  place  as  in 
hydrodynamics. 

BOLTZMANN  says  this  "  normal  "  state  is  permanent  (stationary) 
for  given  external  conditions  because  magnitude  H  does  not 
vary;  such  a  normal  state  has  many  configurations,  but  all  agree 
in  having  same  number  of  complexions. 

Also,  "  MAXWELL'S  Velocity  Distribution  is  not  a  state  which 
assigns  to  each  molecule  a  particular  place  (locus)  and  a  particular 
velocity,  which  are  reached  say  by  the  locus  and  velocity  of  each 
molecule  asymptotically  approaching  said  assigned  locus  and 
velocity.  With  &  finite  number  of  molecules  MAXWELL'S  state  will 
never  be  exactly  but  only  approximately  realized.  MAXWELL'S 
velocity  is  not  a  singular  one  which  is  confronted  by  an  immense 
number  of  non-Maxwellian  velocity  distributions.  On  the  con- 
trary, among  the  immense  number  of  possible  velocity  distributions 
by  far  the  greater  number  possess  the  characteristics  of  the 
MAXWELL  velocity  distribution." 

MAX  PLANCK  (Festschrift,  p.  113)  lucidly  dwells  on  thermal 
equilibrium,  entropy  and  temperature,  as  follows: 

"  The  mechanical  significance  of  the  temperature  idea  is 
most  closely  connected  with  the  mechanical  significance  of  entropy, 


AND  OF  THE  SECOND  LAW  53 

for  the  two  are  connected  by  TdS=dQ.     By  answering  one  of 
these  questions  we  at  the  same  time  settle  the  other." 

In  the  earlier  days  interest  was  naturally  centered  in  the 
directly  measurable  magnitude  temperature  and  entropy  appeared 
as  a  more  complicated  idea  which  was  to  be  derived  from  the 
former.  Nowadays  this  relation  is  rather  reversed  and  the  prime 
question  is  to  first  explain  entropy  mechanically  and  this  will 
then  define  temperature.  The  reason  for  this  change  of  attitude 
is  that  in  all  such  explanatory  efforts  to  present  Thermodynamics 
mechanically  and  give  temperature  a  complete  mechanical  definition 
it  is  necessary  to  come  back  to  the  peculiarities  of  "  thermal  equi- 
librium." But  the  full  significance  of  this  equilibrium  conception 
is.  only  to  be  reached  from  the  standpoint  of  irreversibility.  For 
thermal  equilibrium  can  only  be  defined  as  the  final  state  toward 
which  all  irreversible  processes  strive.  Tn  this  way  the  question 
as  to  temperature  leads  necessarily  to  the  nature  of  irreversibility 
and  this  in  turn  is  solely  founded  on  the  existence  of  the  entropy 
function.  This  magnitude  is  therefore  the  primary,  general 
conception  which  is  significant  for  all  kinds  of  states  and  changes 
of  state,  while  temperature  emerges  from  this  with  the  help  of 
the  special  condition  of  thermal  equilibrium,  in  which  condition 
the  entropy  attains  its  maximum. 

SECTION  B 

SIMPLE  ANALYTICAL  EXPRESSION  FOR  DEPENDENCE  OF  ENTROPY 

ON  PROBABLITY 

Here  also  we  will  dispense  with  a  full  proof  and  content  our- 
selves with  the  main  steps  which  lead  to  the  desired  expression. 
We  will  follow  PLANCK'S  elegant  presentation  on  pp.  136-148  of 
his  Warmestrahlung.  On  p.  22  we  have  dwelt  on  the  usefulness 
and  the  necessity  for  the  probability  idea  in  general  physics  and 
in  this  particular  case.  We  can  start,  therefore,  with  PLANCK'S 
theorem : 


54  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

"  The  entropy  of  a  physical  system  in  a  particular  state  depends 
solely  on  the  probablity  of  this  state." 

No  rigorous  proof  is  here  attempted,  nor  any  numerical  com- 
putations; for  present  purposes  it  will  suffice  to  fix  in  a  general 
way  the  kind  of  dependence  of  entropy  on  probability. 

Let  S  designate  the  entropy  and  W  the  probability  of  a 
physical  system  in  a  particular  state,  then  the  above  theorem 
enunciates  that 

S=f(W),   ........     (8) 

where  f(W)  signifies  a  universal  function  of  the  argument  W. 
Now,  however  W  may  be  defined  we  can  certainly  infer  from  the 
Calculus  of  Probabilities  that  the  probability  of  a  system,  com- 
posed of  two  entirely  independent  systems,  is  equal  to  the  product 
of  the  separate  probabilities  of  the  individual  systems.  For 
example,  if  we  take  for  the  first  sys^n  any  terrestrial  body  what- 
ever and  for  the  second  system  any  hollow  space  on  Sirius,  which 
is  traversed  by  radiations,  then  the  probability  W,  that  simultane- 
ously the  terrestrial  body  will  be  in  a  particular  state  i  and  said 
radiation  in  a  particular  state  2,  will  be  given  by 


(9) 


where  Wi,  W2  respectively  represent  the  separate  probabilities 
of  said  two  states.  Now  let  Si,  S2  respectively  represent  the 
entropies  of  the  separate  systems  corresponding  to  said  states 
i  and  2,  then  according  to  Eq.  8,  we  have 

«Si=/OFi),      S2=f(W2). 

But,  according  to  the  Second  Law  of  Thermodynamics,  the  total 
entropy  of  two  independent  systems  is  5=6*1+52,  and  conse- 
quently according  to  (8)  and  (9), 

^ 


AND  OF   THE   SECOND  LAW  55 

From  this  functional'  equation  /  may  be  determined.  After 
successive  differentiation  there  is  obtained  a  differential  equation 
of  the  second  order  and  its  general  integral  is 

f(W)  =k  log  W+ constant 
or  S=k  log  W+ constant,       ....     (10) 

which  determines  the  general  dependence  of  entropy  on  proba- 
bility. The  universal  integration  constant  k  is  the  same  for  a 
terrestrial  system  as  for  a  cosmical  system,  and  when  its  numerical 
value  is  known  for  either  system  it  will  be  known  for  the  other; 
indeed,  this  constant  k  is  the  same  for  physically  unlike  systems, 
as  above,  where  concurrence  between  a  molecular  and  a  radiating 
system  was  assumed.  The  last,  additive,  constant  has  no  physical 
significance  because  entropy  has  an  arbitrary  additive  constant 
and  therefore  this  constant  in  (10)  may  be  omitted  at  pleasure. 

Relation  (10)  contains  a  general  method  of  computing  the 
entropy  S  from  probability  considerations.  But  the  relation 
becomes  of  practical  value  only  when  the  magnitude  W  of  the 
probability  of  a  system  for  a  certain  state  can  be  given  numerically. 
The  most  general  and  precise  definition  of  this  magnitude  is  an 
important  physical  problem  and  first  of  all  demands  closer  insight 
into  the  details  of  what  constitutes  the  "  state  "  of  a  physical  system. 
[This  has  been  adequately  done  in  the  earlier  part  of  this  presen- 
tation. Later  on  pp.  27,  28,  permutation  considerations  led  us 
to  define  the  probability  IF  of  a  state  as  the  number  of  com- 
plexions included  in  the  given  state.] 


56  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 


SECTION  C 

DETERMINATION  OF  A  PRECISE,  NUMERICAL,  EXPRESSION  FOR  THE 
ENTROPY  OF  ANY  PHYSICAL  CONFIGURATION 

PLANCK  modestly  says  that  everything  essential  in  the  deter- 
mination of  this  expression  has  been  done  by  L.  BOLTZMANN,  in 
his  wide  range  of  physical  investigation.  PLANCK'S  discussion, 
however,  is  so  compact  and  lucid  that  it  is  best  suited  for  our 
purpose.  Keeping  this  purpose  in  mind  we  will  here  also  abbre- 
viate by  dispensing  with  parts  of  PLANCK'S  fuller  proof  and  content 
ourselves  with  the  main  steps  which  lead  to  the  desired  expression. 
These  main  steps  are  as  follows: 

(a)  Determination     of     the     general     expression     for     the 

f  Probability  or  1 

\  XT       .          .     .       TT7  \  of  a  given  physical  configuration  of 

[  No.  of  complexions  W  J 

a  known  macroscopic  state; 

(b)  Determination  of  the  general  expression  for  the  Entropy  5 
of  a  given  physical  configuration  of  a  known  macroscopic 
state; 

(c)  Special  case  of  (b)  namely,  expression  for  the  Entropy  5  of 
the  state  of  thermal  equilibrium  of  a  monatomic  gas; 

(d)  Confirmation,  by  equating  this  value  of  S  with  that  found 
thermodynamically  and  then  deriving  well-known  results. 

(e)  PLANCK'S  conversion  of  the  expressions  of  (b)  and  (c)  into 
more   precise    ones    by    finding    numerical    values    of    k 
in  C.  G.  S.  units;  in  F.  P.  S.  units; 

(f)  Determination  of  the  dimensions  of  the  universal  constant  k 
and  therefore  also  of  entropy  in  general. 


AND   OF  THE  SECOND  LAW  57 


Step  a 

Determination  of  the  Number  of  Complexions  of  a  given  Physical 
Configuration  of  a  Known  Macrostate 

We  will,  for  simplicity's  sake,  consider  here  an  ideal  gas  in  a 
given  macrostate  and  consisting  of  TV-like,  monatomic,  molecules. 
By  generalizing  the  meaning  of  our  co-ordinates,  the  following 
presentation  could  be  made  equally  applicable  to  the  more  general 
case  of  Physics  contemplated  under  this  heading. 

Of  course  we  must  here  have  clearly  in  mind  what  is  meant 
by  the  state  of  a  gas.  For  this  we  may  refer  to  p.  10  (lines  13  to  24) 
and  to  p.  19  (lines  8  to  24).  The  conditions  there  imposed 
are  all  fulfilled  if  we  suppose  the  state  given  in  such  a  way  that 
we  know:  (i)  The  number  of  molecules  in  any  macroscopically 
small  space  (volume  element);  and  (2)  the  number  of  molecules 
which  lie  in  a  certain  macroscopically  small  velocity  region  (soon 
to  be  more  fully  described).  To  have  the  Calculus  of  Probabilities 
applicable,  each  of  the  tiny  regions  contemplated  under  (i)  and  (2) 
must  still  contain  a  large  number  of  molecules  and  their  motions 
must  besides  have  all  the  features  of  haphazard  detailed  on  pp. 
25,  26;  all  this  is  necessary  in  order  that  the  contemplated 
motions  may  possess  all  the  characteristics  of  "  elementary  chaos." 

Before  proceeding  further  on  our  main  line,  we  will  define 
more  fully  what  is  meant  by  the  two  elementary  regions  in  which 
lie  respectively  the  molecules  and  their  velocity  ends.  After 
this  has  been  done  we  will,  for  convenience,  combine  these  two 
regions  into  a  fictitious  elementary  region,  say,  a  six-dimensional 
one. 

First  there  is  the  volume  element  dx-dy-dz^dV,  in  which 

(x  and  x  +  dx  T 
y  and  y  +  dy  \ 
z  and    z  +  dz  J 
is  located;    this  element  can  be  conceived  as  a  parallelopipedon 


58  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

whose  edges  are  parallel  to  the  co-ordinate  axes  ;  this  is  the  simplest 
of  the  elementary  regions  here  to  be  considered.  To  conceive 
of  the  elementary  region  ds-dy-d^  containing  the  velocity  ends  of 
the  molecules,  let  us  suppose  any  origin  O  for  velocities  in  a  unit 
volume  and  from  this  as  a  pole  lay  off  as  vectors  the  molecular 
velocities  lying  between  the  limits, 


and 

and  Tj+dy,         ......     (n) 

and  ^ 


where  £,  ij,  £  are  the  components  of  said  velocities  parallel  to  the 
respective  co-ordinate  axes.  Then,  under  the  velocity  limitations 
imposed,  the  end  of  the  velocity  of  each  such  molecule  will  lie  in 
the  elementary  parallelepiped  dZ-dy-d^,  one  vertex  of  this 
parallelepiped  having  of  course  the  co-ordinates  £,  y,  £.  This 
parallelepiped  can  be  regarded  as  a  constructed  volume  within 
which  the  velocity  end  must  lie.  We  have  therefore  here  two 
elementary  volumes  dx-dy-  dz  and  d^-dy-d^  which  are  independent 
of  each  other,.  Now  remembering  that  the  probability  of  any 
properly  endowed  molecule  being  found  in  one  of  these  volumes 
is  in  each  case  equal  to  the  number  of  molecules  belonging  or 
corresponding  to  the  volume  considered.  Assuming,  for  the 
moment,  an  equal  distribution  of  molecules  and  velocities  through- 
out the  whole  volume,  we  may  say  that  the  number  of  molecules 
occurring,  in  each  of  the  said  elementary  volumes,  is  proportional 
to  their  respective  sizes;  this  is  here  equivalent  to  saying  that 
the  probability  of  any  molecule  thus  occurring  in  said  elementary 
volumes  is  proportional  to  their  respective  sizes.  Having  stated 
the  probability  of  each  contemplated  occurrence,  we  can  now  say 
the  probability  of  these  two  events  concurring  is  equal  to  the  product 
of  the  probabilities  of  said  two  separate  occurrences.  Moreover, 
as  the  probability  of  each  occurrence  is  proportional  to  the  size 


AND   OF  THE  SECOND   LAW  59 

of  its  own  elementary  volume,  the  product  of  said  probabilities 
will  likewise  be  proportional  to  the  product 

dx-dy-dz-dZ-drj-d^^a (12) 

of  the  two  elementary  volumes.  Here  a  can  be  regarded  as  a  sort 
of  fictitious  volume  or  region,  constructed,  say,  in  a  six-dimensional 
space.1 

The  extent  of  such  an  elementary  region  is  very  minute  in 
comparison  with  the  total  space  under  consideration,  but  still 
it  must  be  conceived  as  sufficiently  large  to  embrace  many  mole- 
cules, otherwise  its  state  would  not  be  one  of  "  elementary  chaos." 
On  account  of  the  equivalence  here  of  probability  and  number 
of  concurring  molecules,  we  may  for  the  present  say  that  the 
number  of  the  latter  is  proportional  to  the  magnitude  of  this 
elementary  region  a.  But  before  we  proceed  further  this  last 
statement  must  be  subjected  to  a  correction,  for  we  temporarily 
assumed  above  that  there  was  an  equal  distribution  of  molecules 
and  velocities  throughout  the  whole  volume.  Now  at  the  start, 
in  defining  the  contemplated  state,  it  was  distinctly  announced 
that  there  was  an  unequal  distribution  of  such  elementary  con- 
ditions, the  law  of  their  distribution  being  given  by  the  known 
number  of  molecules  in  each  elementary  volume  dV  and  in  each 
constructed  elementary  velocity  volume  d^-dy-d^.  This  correction 
is  effected  by  the  introduction  of  a  finite  proportionality  factor, 

/(*,?,«,£,  7,  Q,    ......     (13) 

which  can  be  any  given  function  of  the  location  and  velocity 
co-ordinates,  so  long  as  it  fulfills  the  one  condition  (put  in  abbre- 
viated form), 

S/-<T  =  ,V, (14) 

where  N  =  the  total  number  of  molecules  of  the  gas. 

1  Such  a  fictitious  space  does  not  occur  in  the  proof  of  MAXWELL'S  distribution, 
because  there  conditions  are  simpler.  See  foot-note  to  p.  49. 


60  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

Strictly  speaking,  the  expression  a  for  the  fictitious  elementary 
region  a,  formed  by  the  product  of  dV  and  the  constructed- 
volume  element  dZ-dy-d^,  should  be  replaced  by  the  expression 
m3a,  where  m  is  the  mass  of  a  molecule.  The  reason  for  this 
substitution  is  found  in  the  fact  that  the  magnitude  of  the  con- 
structed-volume  element  dZ-dy-dt,  varies  with  time  due  to  the 
variation  of  velocities  effected  by  molecular  collisions.  Now 
this  variation  of  magnitude  is  not  permissible  with  the  probability 
considerations  which  here  obtain.  For  the  probability  of  a  state 
which  necessarily  follows  from  another  state  must  be  like  that 
of  the  latter.  As  the  momenta  after  collision  are  the  same  as 
before  collision,  we  have  now  in  the  momenta,  co-ordinates  which 
do  not  vary  with  time  like  their  constituent  velocities.  Therefore 
if  we  substitute  in  (12)  for  the  velocities  £, >?,  £  their  corresponding 
momenta,  the  variation  with  time  of  the  constructed-volume  will 
cease  and  the  objection  cited  will  no  longer  be  a  valid  one. 

Now  let  us  take  up  the  determination  of  the  number  of  com- 
plexions W  in  the  given  state.  For  this  purpose  think  of  this 
whole  state  as  represented  by  the  sum  total  of  all  these  equal 
elementary  regions  mza\  for  convenience  of  reference  let  us  call 
this  whole  state  the  "  state-region."  The  probability  that  a  par- 
ticular molecule  will  belong  to  a  particular  elementary  region 
is  equally  great  for  all  the  elementary  regions.  Let  P  represent 
the  number  of  these  equal  elementary  regions.  Now  we  will 
proceed  with  the  help  of  a  parallel  case.  Let  us  think  of  as  many 
dice  N  as  there  are  molecules  and  let  each  die  be  provided  with 
P  faces.  On  each  of  these  faces  we  will  write  in  their  order  the 
digits  i,  2,  3,  .  .  .  P,  so  that  each  of  the  P  faces  will  designate 
a  particular  elementary  region.  Then  each  throw  of  the  N 
dice  will  result  in  representing  a  particular  state  of  the  gas,  because 
the  number  of  dice  which  show  uppermost  a  particular  digit 
will  give  the  number  of  molecules  belonging  to  the  elementary 
region  represented  by  said  digit.  In  this  parallel  case  each  die  is 
equally  likely  to  show  up  any  one  of  the  digits  i  to  P,  corresponding 


AND  OF  THE  SECOND  LAW  61 

i 

to  the  circumstance  that  each  individual  molecule  is  equally 
likely  to  belong  to  any  one  of  the  elementary  regions.  The 
desired  probability  W  of  the  given  state  of  the  molecules  corre- 
sponds therefore  to  the  number  of  different  kinds  of  throws  (com- 
plexions), by  which  the  given  distribution  /  can  be  realized. 
For  example,  if  we  take  N=io  molecules  (dice)  and  P  =  6 
elementary  regions  (dice  faces),  and  assume  that  the  state  is  so 
given  that  it  is  represented  by: 

3  molecules  in  elementary  region  i 

4  "  "  "  2 

0  "  "  "  3 

1  ll  "  "  4 
o  "  "  "  5 

2  "  "  "  6 

Then  this  state  can  be  realized  by  one  throw,  in  which  the  10 
dice  show  the  following  digits : 

ist   ad   3d   4th  sth   6th   ?th   8th   gth   loth  die 

the          2621126214    digit       (15) 

Under  each  of  the  10  dice  stands  the  digit  shown  uppermost 
in  the  throw.  In  fact,  we  see 

3  dice  with  digit  i 

4  "         "          2 

0  "        "         3 

1  "  4 
o       "        "         5 

2  6 

In  like  manner  the  same  state  can  be  realized  by  many  other 
such  complexions.  The  desired  number  of  all  possible  complex- 
ions can  be  found  by  considering  the  digit  row  designated  above 
by  (15).  For,  since  the  number  of  molecules  (dice)  is  given, 


62  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

the  digit  row  will  have  a  particular  number  of  places  (io  =  .ZV). 
Besides,  since  the  number  of  molecules  belonging  to  each  elemen- 
tary space  is  given,  each  digit  will  occur  equally  often  in  the  row 
in  all  permissible  complexions.  Moreover  every  change  in  digit 
arrangement  effects  a  new  complexion.  The  number  of  the 
possible  complexions  or  the  probability  W  of  the  given  state  is 
therefore  equal,  under  the  conditions  specified,  to  the  possible 
"  permutations  with  repetition."  1  In  the  simple  example  chosen 
we  have  for  such  permutation,  according  to  a  known  formula, 

10! 

12,600. 


3!  4!  o!  i  !  o!  2 
Consequently  in  the  general  case,  we  have 

N\ 


where  the  sign  n  signifies  the  product  extended  over  all  the  P 
elementary  regions. 

Result  contained  in  (16)  is  equally  true  for  any  other  physical 
system,  say,  one  involving  radiant  energy.  For  the  conditions 
and  the  variables  are  similar  to  those  of  the  molecular  system 
just  employed.  The  chosen  model,  the  dice  system,  which  served 
as  an  easily  conceived  parallel  case,  would  be  equally  serviceable 
in  dealing  with  the  elements  of  radiation. 

1  Compare  with  pp.  27  and  28  where  this  permutation  process  is  discussed  some- 
what fully. 


AND  OF  THE  SECOND  LAW  63 


Step  b 

Determination  of  a  General  Expression  for  the  Entropy  S  of  any 
given  Natural  State 

This  step  is  an  easy  one.  We  have  in  Eq.  (10)  the  relation 
expressing  the  universal  dependence  of  entropy  S  on  probability 
W.  Substituting  and  writing  out  the  logarithm  of  the  quotient 
given  in  (16),  we  have 

S  =  HogAn-£Slog(/.<7)!.     ....     (17) 

The  summation  S  must  be  extended  over  all  the  elementary 
regions  a.  With  the  help  of  STIRLING'S  formula,  and  remembering 
that  both  o  and  N=^f-a  are  constant  for  all  changes  of  state, 
the  above  expression  (17)  is  reduced  to  the  form 

Entropy  5= constant— ££/-log/-<7.      .     .     ...    (18) 

This  magnitude  5  is  numerically  the  same  as  H  for  which  BOLTZ- 
MANN  proved  that  it  changed  in  a  one-sided  way  in  all  changes 
of  state.  We  must  bear  in  mind  too  that  function  /  represents, 
for  every  state  of  the  gas,  the  given  space  and  velocity  distribution 
of  the  gas  molecules.  The  permanent,  stationary,  state  of  the 
gas  known  as  thermal  equilibrium  is  only  a  special  case  of  the 
general  case  (18),  this  special  case  being  widely  known  as  MAX- 
WELL'S Law  of  Distribution  of  Velocities. 


Step  c 

Special  Case  of  (b),  Namely,  Determination  of  Entropy  S  for  the 
Thermal  Equilibrium  of  a  Monatomic  Gas 

This  case  PLANCK  derives  very  easily  from  the  general  case 
represented  by  (18).  As  the  desired  result  has  already  been 
found  in  another  way  in  pp.  48-53  when  dealing  with  MAXWELL'S 


64  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

Law  (of  distribution  of  molecular  velocities),  we  will  not  repeat 
PLANCK'S  derivation  of  the  law  from  (18).  It  will  suffice  here  to 
give  the  results:  The  law  of  distribution  is  given  by  function 


where  a  and  /?  are  constants  and  U  the  total  energy.  As  this  expres- 
sion for  function/is  free  from  all  location  co-ordinates  x,  y,  z,  we  see 
that  this  state  of  thermal  equilibrium  is  independent  of  these 
space  co-ordinates  and  conclude  that  in  this  state  the  molecules 
are  uniformly  distributed  in  space;  only  the  velocities  are  variously 
distributed,  all  of  which  accords  with  the  earlier  presentation. 
Substituting  the  results  of  (19)  in  the  general  equation  (18)  there 
results  for  the  entropy  5  of  the  state  of  equilibrium  of  a  monatomic 

§as'  S  =  constant  +  kN($  log  U+logV).     .     .     .     (20) 

To  make  Eq.  (20)  practically  serviceable  we  need  to  know  the 
constants  k  and  N  and  they  will  be  found  later  on. 

Step  d 

Confirmation,  by  Equating  this  Probability  Value  of  S  with  that 
found  Thermodynamically  and  Securing  well-known  Results 

We  know  from  Thermodynamics  that  the  change  of  entropy 
is  denned  in  a  perfectly  general  way  (for  physical  changes)  1  by 

__     dU  +  pdV 
dS  =     —^  -  .......     (21) 

Deriving  the  partial  differential  coefficients  and  making  use 
of  (20),  there  follows:  kNT  RnT 

P=—y-=~>  ......     (22) 

1  This  differential  equation  is  valid  only  for  changes  of  temperature  and  volume 
of  the  body  but  not  for  its  changes  of  mass  and  of  chemical  composition,  for  in 
in  defining  entropy  nothing  was  said  of  these  latter  changes. 


AND  OF  THE  SECOND   LAW  65 

where  n=  number  of  gram-molecules  (referred  to  O2=32g)  and 

erg 
R  =8.315  (io7)  -7  —  =  absolute  gas  constant  [1545  in  F.P.S.  system] 

Here  the  first  of  Eqs.  (22),  represents  the  combined  laws  of  BOYLE, 
GAY-LUSSAC,  and  AVOGADRO.  We  get  besides  from  the  equating 
of  (20)  and  (21),  the  additional  relations, 


(23) 


where  mechanical  equivalent  A  =4.i9(io5)  — p  C.G.S.  system. 
From  this  follows 

^=3-°i     CP  =  $>     and     ^  =  7, (24) 

cv     3 

as  is  known  for  monatomic  gases. 

Furthermore,  we  find  for  the  mean  kinetic  energy  L  of  a 
molecule 

L  =  ~=^-kT.  (25) 

N     2 

We  also  have 

n      w  wt.  of  a  molecule 

&=—-=  —  =  — — —  =  const,  for  all  gases.  (26) 

N     m     molecular  wt.  of  a  molecule 

With  the  help  of  the  specific  heats  and  the  characteristic  equation 
of  the  gas,  the  whole  thermodynamic  behavior  of  the  gas  is  disclosed. 
All  this  has  resulted  from  the  ^identification  of  the  mechanical  and 
thermodynamic  expressions  for  entropy  and  is  an  indication  of 
the  fruitfulness  of  the  method  pursued.  PLANCK  also  shows  that 
this  method  leads  to  the  finding  of  results  heretofore  unknown. 


66  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 


Step  e 

Conversion  of  the  General  Expressions  in  (b)  and  (c)  info  more 
Precise  ones  by  Finding  and  Inserting  the  Numerical  Value  oj 
the  Universal  Constant  k  ;  Some  of  the  Results 

From  the  consideration  of  certain  phenomena  of  radiation 
PLANCK  found 


erg 

~16     *  absolute  c  G-  s- 


"          in  F-  P-  S* 


where  a  and  b  are  constants  found  by  experiment  while  a  and  ft 
are  exactly  known  values,  mathematically  derived.  The  present 
accuracy  of  (27)  therefore  rests  on  the  accuracy  of  the  experiments 
from  which  a  and  b  were  found.  In  discussing  Eq.  (10)  it  was 
pointed  out  that  k  was  a  universal  constant,  applicable  to  all 
physical  systems  and  consequently  may  be  used  for  the  molecular 
configurations  mainly  considered  in  this  presentation.  But  before 
introducing  numerical  value  of  k  in  the  general  expressions  con- 
tained under  headings  b  and  c,  we  will  add  other  numerical 
values  of  interest. 

PLANCK  gives  2.76(io19)  =number  of  molecules  in  i  ccm.  of  an 
ideal  gas  at  freezing-point   (o°  C.)   and   atmospheric  pressure; 

N 
he  also  gives  for  the  ratio  —  =6.i75(io23)  =number  of  molecules 

per  m  grams;  the  corresponding  numbers  in  English  units  are, 
approximately,  782  (zo23)  =  number  of  molecules  in  one  cubic  foot 

J    N       2.80(I026) 

of  an  ideal  gas  and  —  =  —    -  =  number  of  molecules  in  one 
n  m 

pound  of  an  ideal  gas.  Assuming  air  to  be  an  ideal  gas  and  its 
"  apparent  "  molecular  weight  about  28.88,  the  number  of  mole- 


AND  OF  THE  SECOND  LAW  67 

2.8o 
cules  in  one  pound  of  air  would  be  (lo26)  =o.g7(io25). 

Substituting  the  numerical  value  of  universal  constant  k  in 
Eq.  (10)  we  get  Eq.  (28). 
For  C.  G.  S.  system,  Entropy  of  any  natural  state,  Eq.  (28)  is 

5  =  i.346(io~16)  loge  (W)  =  i.  346(10  ~16)  loge  (No.  of  complexions). 
For  F.  P.  S.  system,  Entropy  of  any  natural  state,  Eq.  (28)  is 
5  =  5.5o(io~24)  loge  (W)  =5.5o(io~24)  loge  (No.  of  complexions.) 

To  each  of  these  may  be  added  an  arbitrary  constant.  In 
Eq.  (20)  we  may  substitute  directly  the  equivalent  of  the  product 
kN  found  from  Eq.  (22),  and  then  get  for  the  entropy  S  of  a 
monatomic  gas  in  the  state  of  thermal  equilibrium, 


(29) 


When  the  volume  V  is  known  we  can  now  readily  find  N 
and  then  kN  numerically,  and  place  this  number  as  a  coefficient 
in  Eq.  (20). 

Stepf 
Determination  of  the  Dimensions  of  k  or  of  the  Entropy  S 

It  is  at  once  evident  from  an  inspection  of  the  perfectly  general 
Eq.  (10)  that  the  dimensions  of  Entropy  5  depend  solely  on  those 
of  the  universal  constant  k.  The  relation  given  in  Eq.  (21)  shows 
at  once  that  dimensions  of  dS  depend  upon  the  quotient  found 
by  dividing  energy  by  temperature  and  the  relations  given  in 
Eqs.  (22)  and  (25)  that  the  dimensions  of  constant  k  also  depend 
on  this  same  quotient.  The  dimensions  of  Entropy  S  and  of  con- 
stant k  are  therefore  identical  and  this  might  suffice  to  show  that 
here  neither  S  nor  k  is  to  be  regarded  as  a  mere  ratio  or  abstract 


68  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

number.  A  word  further  in  this  connection  may,  however,  be 
helpful.  In  reversible  processes  we  have  the  well-known  relation 
dQ  =  TdS.  To  simplify  matters,  let  us  suppose  heat  dQ  supplied 
while  volume  is  kept  constant,  then  dQ  =  cvdT=  TdS  or 

^=^r'.  *  .....  (30) 


Here  Entropy  S  has  the  same  dimensions  as  cv;  now  in  the 
relation  dQ  =  cjlT  if  we  regard  cv  as  an  abstract  number  then, 
in  order  that  the  equation  shall  be  homogeneous  the  factor  (dT) 
must  represent  heat  energy  like  dQ,  and  this  is  sometimes  done; 

7/T1 

hi  such  case  (if  T  retains  its  ordinary  meaning)  the  quotient  -=7 

in  Eq.  (30)  is  no  longer  a  mere  ratio  or  abstract  number,  but  a 
quotient  of  the  dimensions  of  energy  divided  by  temperature. 

dQ 
On  the  other  hand,  if  cv=jf  be  regarded  as  of  the  dimensions 

of  the  quotient  of  energy  divided  by  temperature,  then  we  may 

/IT* 

consider  -~  in  (30)  as  an  abstract  number  or  ratio  and  dS  of  the 

same  dimensions  as  Cv.  When  an  absolute  system  of  units  is 
employed,  which  possesses  as  one  of  its  features  the  expression  of 
temperature  in  units  of  energy,  then  k,  S  and  cv  will  all  be  mere 
ratios  or  abstract  numbers.1 

iSee  C.  V,  BURTON'S  article  in  Philosophical  Transactions,  Vol.  23-24,  1887. 


AND  OF  THE  SECOND  LAW  69 


PART   III 

PHYSICAL  INTERPRETATIONS 
SECTION  A 

OF  THE  SIMPLE  REVERSIBLE  OPERATIONS  IN  THERMODYNAMICS 

Change  under  Constant  Volume 

WE  found  above  that  the  entropy  of  a  state  was  precisely 
defined  in  a  physical  way  by  the  number  of  complexions  of  that 
state.  Now  let  us  see  what  happens  to  this  number  of  com- 
plexions when  an  ideal  gas  experiences  some  of  the  simpler 
changes,  of  a  reversible  (non-cyclical)  character.  We  will  begin 
with  the  case  in  which  the  volume  of  the  gas  remains  constant 
while  its  temperature  rises,  the  final  state  of  the  gas  having  a 
higher  temperature  than  its  initial  state. 

We  see  from  Eq.  (7),  p.  51,  that  c  grows  and  from  Eq.  (4), 
p.  50,  that  C  diminishes.  MAXWELL'S  Law,  given  by  Eq.  (5), 

p.  50,  shows  for  a  given  velocity  —  that  the  number  -r-  of  molecules 

c  ap 

possessing  the  given  velocity  is  less  in  the  final  state  than  it  was 
in  the  initial  state,  and  as  the  total  number  n  of  molecules  in  the 
gas  is  unchanged,  there  will  be  a  greater  variety  of  velocities  in 
the  final  state.  This  makes  the  number  of  possible  permutations 
greater  in  the  final  state,  thus  increasing  the  number  of  com- 
plexions; consequently,  as  entropy  varies  with  the  logarithm  of 
the  number  of  complexions,  we  see  that  the  entropy  of  the  final 
state  is  greater  than  in  the  initial  state  and  this  agrees  with 
experience. 


70  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 


Isobaric  Change 

Next  we  interpret  how  the  number  of  complexions  are  affected 
by  isobaric  change  during  a  reversible  process,  again  assuming 
that  the  temperature  in  the  final  state  is  greater  than  in  the 
initial  one.  Here  the  steps  and  the  conclusion  are  exactly  the 
same  as  in  the  preceding  case.  In  both  cases  just  the  opposite 
result  is  reached  when  there  is  a  fall  in  temperature. 

As  the  pv  diagram  contains  the  co-ordinates  p,  v,  and  repre- 
sents mainly  the  mechanical  changes  in  the  body  under  considera- 
tion, we  can,  by  suitable  combination,  similarly  interpret  any 
other  reversible  change  of  state  represented  in  this  pv  diagram. 

Isothermal  Change 

However,  because  of  its  general  importance  and  because  of 
its  bearing  on  the  temperature-entropy  diagram,  we  will  here  also 
tell,  in  the  same  physical  terms,  what  happens  when  our  ideal 
gas  undergoes  isothermal  change  with  increase  of  volume.  As 
the  temperature  in  the  final  state  is  equal  to  that  in  the  initial 
one,  the  quantity  [^2]=fc2  does  not  change  and  therefore  C  does 

not  change  nor  (see  Eq.  (5),  p.  50)  does  the  number  -j-  of  mole- 
cules possessing  the  velocity  —  change.  The  variety  of  velocities 

Cr 

in  the  final  state  is  therefore  the  same  as  in  the  initial  state  and 
does  not  at  all  contribute  to  that  necessary  increase  in  the  num- 
ber of  complexions  (configurations)  for  which  we  are  looking. 

The  direction  of  the  velocity  of  a  molecule  would  be  another 
variety  element,  but  as  the  final  volume  evidently  possesses  as 
many  velocity  directions  as  the  initial  volume,  this  element  or 
co-ordinate  will  not  contribute  to  increased  complexity  in  the 
final  state.  But,  as  the  volume  has  increased,  the  final  state 
will  contain  more  unit  volumes  (and  these  can  be  taken  as  small 


AND   OF  THE  SECOND  LAW  71 

as  we  please)  than  the  initial  state.  As  it  is  here  equally  likely 
that  a  particular  molecule  will  be  found  in  any  one  of  these  unit 
volumes,  it  is  evident  that  the  increase  of  volume  will  add  increased 
variety  to  the  location  or  configuration  of  the  molecules  and  by 
indulging  in  the  swapping  of  places  inherent  in  the  production  of 
complexions,  we  see  that  said  increment  of  volume  will  make 
the  number  of  complexions  in  the  final  state  greater  than  in  the 
initial  state,  which  in  turn  means  that  the  entropy  in  the  final 
state  is  also  greater.  This  accords  with  experience,  but  it  can 
also  be  seen  from  the  formula 

Entropy  S=constant+£Npogf  £7+log  F],  .    .     (31) 

derived  by  PLANCK  (p.  63  of  Vorl.  ii.  Theor.  Physik),  from 
probability  considerations,  for  the  state  of  thermal  equilibrium. 
Here  k  is  the  universal  constant  (see  p.  66)  and  the  other  terms 
have  the  same  meaning  as  before. 

Isentropic  Change 

The  last  reversible  process,  to  be  here  physically  interpreted, 
is  isentropic  change  from  the  initial  state  of  thermal  equilibrium 
to  its  final  state.  Evidently  only  the  physical  elements  under- 
lying the  bracketed  term  in  Eq.  (31)  need  to  be  considered. 

As  we  are  considering  isentropic  change  (dS=o),  it  does  not 
make  any  difference  whether  on  the  one  hand  we  think  of  this 
isentropic  change  as  accompanied  by  an  increase  in  temperature 
and  decrease  in  volume,  or  on  the  other  hand  think  of  said  change 
as  taking  place  with  decrease  of  temperature  and  increase  of 
volume.  Suppose  we  assume  the  latter  kind  of  change.  Then 
from  what  has  preceded  we  know  that  increase  of  volume  by 
itself  would  increase  the  number  of  complexions  of  the  final 
state,  also,  from  what  has  gone  before,  we  know  that  the  drop 
in  temperature  by  itself  will  lead  to  decrease  in  the  number  of 
complexions  in  the  final  state.  These  two  influences  acting 


72  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

simultaneously  therefore  tend  to  neutralize  each  other  and  if 
they  exist  in  the  proper  ratio,  derivable  from  the  bracketed  quantity 
in  Eq.  (31),  they  will  completely  balance  each  other  and  produce 
no  change  whatever  in  the  number  of  complexions  while  passing 
from  the  initial  to  the  final  state  of  equilibrium,  i.e.,  will  produce 
no  change  whatever  m  the  entropy  of  the  gas  under  consideration. 
In  isentropic  change  Nature  has  no  preference  for  its  various 
states. 

The  temperature-entropy  diagram  considers  mainly  thermal 
changes,  and  as  we  have  considered  the  influence  of  both  of  its 
co-ordinates  in  the  number  of  complexions,  we  can  ascertain  by 
proper  combination,  for  any  reversible  change  of  state,  the  corre- 
sponding character  of  the  change  in  the  number  of  complexions. 
It  is  evident,  too,  that  in  the  diagram  any  reversible  change  of 
state  is  equivalent,  so  far  as  the  change  of  entropy  in  the  one  body 
is  concerned,  to  an  isentropic  change  combined  with  an  isothermal 
change,  the  latter  only  affecting  the  result,  so  far  as  change  in 
nttmber  of  complexions  is  concerned. 

SECTION  B 
OF  THE  FUNDAMENTALLY  IRREVERSIBLE  PROCESSES 

If  we  consider  only  heat  and  mechanical  phenomena  and  do 
not  include  electrical  occurrences,  the  irreversible  processes  may 
be  grouped  in  four  classes: 

(a)  The  body  whose  changes  of  state  are  considered  is  in 
contact  with  one  or  more  bodies  whose  temperatures  differ  by  a 
finite  amount   from   its   own.     There  is  here  flow  of  heat  from 
hot  to  cold  and  the  process  is  an  irreversible  one. 

(b)  When  the  body  experiences  friction  which  develops  heat 
it  is  not  possible  to  effect  completely  the  opposite  operation. 

(c)  The  third  group  includes  those  changes  of  state  in  which 
a  body  expands  without  at  the  same  time  developing  an  amount 
of  external  energy  which  is  exactly  equal  to  the  work  of  its  elastic 


AND   OF  THE  SECOND  LAW  73 

forces.  For  example  this  occurs  when  the  pressure  which  a 
body  has  to  overcome  is  essentially  (that  is,  finitely)  less  than 
the  body's  own  internal  tension.  In  such  a  case  it  is  not  possible 
to  bring  said  body  back  to  its  initial  state  by  a  completely  opposite 
procedure.  Examples  of  this  group  are:  steam  escaping  from 
a  high-pressure  boiler,  compressed  air  flowing  into  a  vacuum  tank 
and  a  spring  suddenly  released  from  its  state  of  high  tension. 

(d)  Suppose  two  gases  existing  at  the  same  pressure  and 
temperature  are  on  opposite  sides  of  a  partition;  when  the  par- 
tition is  quickly  removed  the  two  gases  will  diffuse  or  mix.  These 
gases  will  not  unmix  of  themselves  and  the  diffusion  process  is 
an  irreversible  one  and  is  somewhat  like  the  process  considered 
under  (c). 

The  foregoing  facts  and  propositions  have  in  the  main  already 
been  stated  in  this  presentation  and  it  will  be  profitable  to  make 
comparisons  with  the  definition  of  irreversible  and  reversible 
events  given  on  p.  30  and  with  the  examples  on  pp.  31,  32. 

HEAT  CONDUCTION 

The  group  under  head  (a)  represents  the  irreversible  processes 
which  perhaps  occur  most  often,  namely,  the  direct  passage  of 
heat,  by  conduction  or  radiation,  from  a  hot  body  to  a  cold 
body,  here  say  from  a  hot  gas  to  a  cold  gas.  The  former  loses 
in  heat  energy  what  the  latter  gains.  As  radiation  phenomena 
have  very  special  features  of  then*  own  and  for  the  present  may 
be  said  to  be  outside  of  our  selected  province,  we  will  confine 
our  attention  to  heat  conduction  alone.  Moreover,  for  our  present 
purpose,  we  will  suppose  said  flow  or  change  to  take  place  without 
alteration  of  volume  of  either  the  hot  or  the  cold  gas.  Then  will  the 
hot  gas  experience  a  drop  in  temperature  and  the  cold  one  a  rise 
in  temperature.  We  have  already  treated  such  isometric  changes 
and  know  that  the  number  of  complexions  is  thereby  diminished 
in  the  originally  hotter  body  and  increased  in  the  originally  colder 


74  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

one.  If  this  increment  is  greater  than  the  accompanying  decre- 
ment, then  the  final  outcome  of  this  direct  passage  from  hot  to 
cold  is  an  increase  in  the  total  number  of  complexions  of  the 
two  gases.  There  will  then,  by  our  precise  definition,  be  a  con- 
responding  increase  in  the  total  entropy  of  the  two  systems.  It 
is  foreign  to  our  present  purpose  to  prove  in  an  independent, 
purely  mechanical  way,  that  such  excess  does  finally  exist  and 
will  here  content  ourselves  with  the  well-known  and  simple 
thermodynamic  expression  for  this  excess, 


(32) 


where  Q  is  the  heat  energy  thus  directly  transferred  from  the 
hot  to  the  cold  body,  TI  the  absolute  temperature  of  the  hot  body 
and  T2  that  of  the  cold  body. 

THE  WORK  OF  FRICTION  is  CONVERTED  INTO  HEAT 

The  group  under  head  (&)  contains  a  class  of  events  which 
usually  attends,  in  one  form  or  another,  most  natural  phenomena. 

We  will  here  consider  an  interesting  (but  perhaps  too  special) 
case,  namely,  the  experiment  performed  by  W.  THOMSON  and 
JOULE  on  the  flow  of  gas  through  a  porous  plug.  The  plug 
obstructed  the  uniform,  non-conducting,  passage  through  which 
the  gas  was  forced  without  sensibly  changing  its  velocity  of  flow: 
(See  LORENZ'  Technische  Warmelehre,  p.  275).  It  can  easily  be 
shown  (in  L.,  p.  274)  that  with  an  ideally  perfect  gas, 

Final  temperature  T12=-T'i=  initial  temperature. 

As  a  matter  of  fact,  there  was  an  actual  though  slight  drop  in 
temperature  found  to  exist  with  the  most  perfect  gases  available. 
Evidently  the  process  was  a  throttling  one,  reducing  the  larger 
initial  pressure  to  the  smaller  final  one,  which  reduction  was  of 
course  accompanied  by  a  corresponding  increase  in  volume. 


AND   OF  THE  SECOND   LAW  75 

Assuming  that  an  ideally  perfect  gas  was  employed  in  the 
experiment,  and  that  the  final  state  for  our  consideration  is  that 
corresponding  to  its  attainment  of  thermal  equilibrium,  we  see 
that  because  of  the  unchanged  temperature  there  is  here  no  loss 
of  internal  energy,  for  the  work  consumed  by  the  friction  of  the 
porous  plug  is  all  returned  to  the  gas  by  the  heat  developed  by 
said  friction.  Moreover,  the  +  and  —  external  work  in  this 
experiment  also  balance.  Now  although  there  has  been  no  loss 
of  energy  there  has  been  a  growth  of  entropy  corresponding  to 
the  evident  increase  in  the  number  of  complexions.  This  increase 
is  exactly  equal  to  that  found  for  reversible  isothermal  change 
of  state  when  accompanied  by  an  increase  in  volume,  and  the  dis- 
cussion is  therefore  not  repeated  here. 

One  phase  of  the  above  process  is  the  conversion  of  mechanical 
work  into  heat  through  the  medium  of  friction. 

INCREASE  OF  VOLUME  WITHOUT   PERFORMANCE  OF  EXTERNAL 
WORK  BY  ELASTIC  FORCES  OF  THE  GAS 

This  case  of  an  irreversible  process  comes  into  group  c.  We 
will  consider  here  JOULE'S  well-known  experiment  with  the  air 
tanks,  in  which  the  compressed  air,  initially  stored  in  the  one 
tank,  was  allowed  to  discharge  into  the  other  tank  which,  at  the 
start,  contained  only  a  vacuum.  At  the  end  of  the  experiment, 
when  thermal  equilibrium  obtained,  the  temperature  in  the  two, 
now  connected,  tanks  was  the  same  as  originally  existed  in  the 
compressed-air  tank.  Here  of  course  it  is  assumed  that  the  air 
exchanged  no  heat  whatever  with  the  outside. 

As  the  final  state,  like  the  initial  state,  is  in  thermal  equilibrium, 
and  possesses  the  same  temperature,  we  can  ascertain  the  total 
change  in  the  number  of  complexions  as  we  did  when  discussing 
isothermal  and  reversible  changes  and  because  of  the  accompany- 
ing increase  in  the  volume  of  the  air,  find  that  here  as  there  the 
number  of  complexions  has  increased  and  that  therefore  the 
entropy  of  the  air  has  increased  in  this  case. 


76  THE  PHYSICAL   SIGNIFICANCE   OF   ENTROPY 

We  might  rest  satisfied  with  this  conclusion,  but  additional 
light  will  be  shed  on  entropy  significance  if  we  consider  more 
in  detail  the  intermediate  stages  of  this  evidently  irreversible 
process.  The  rush  of  air  from  the  full  to  the  empty  tank  produces 
whirls  and  eddies  of  a  finite  character  and  it  is  only  when  these 
have  subsided,  by  the  conversion  of  the  visible  or  sensible  kinetic 
energy  of  their  particles  into  heat,  that  thermal  equilibrium 
obtains.  But  at  each  intermediate  stage  (while  still  visibly  whirl- 
ing and  eddying)  the  .gas  possesses  entropy,  even  while  in  the 
turbulent  condition.  This  is  clear  from  our  present  physical 
definition  of  entropy,  namely,  the  logarithm  of  the  number  of 
complexions  of  the  state,  for  it  is  evident  that  even  in  this  turbulent 
state  it  possesses  a  certain  number  of  complexions,  however 
difficult  mathematically  it  may  actually  be  to  find  this  number. 
BOLTZMANN  found  an  expression  for  any  condition ;  PLANCK  gave 
it  the  form  of  Eq.  (18),  p.  63, 

En  tropy  =  5*  =  constant -&£/•  log /-<7,    ..."  (33) 

where  ~k  =  1.346  (io~16)  (in  theC.G.S.  system)  is  a  universal  constant, 
function  /  is  the  law  of  distribution  of  the  particles  and  their 
velocity  elements  and  a  =  dx-dy-dz'd£-dr}'dtt  is  a  sort  of  fictitious 
elementary  region  in  a  six-dimensional  space.  From  its  deriva- 
tion and  definition  the  value  given  for  entropy  S  in  Eq.  (33) 
depends  only  on  the  state  of  the  body  at  the  instant  in  question 
and  does  not  at  all  depend  on  its  history  preceding  this  instant. 

The  difference  between  the  value  of  5  for  the  final  state  (say, 
as  given  for  a  gas  by  Eq.  20)  and  the  value  of  S  as  given  by  Eq.  (33) 
for  the  instant,  constitutes  the  driving  motive  which  urges  the 
gas  toward  thermal  equilibrium.  A  similar  difference  or  driving 
motive  is  the  underlying  impelling  cause  of  all  natural  phenomena. 


AND   OF   THE  SECOND   LAW  77 


OF  THE  DIFFUSION  OF  GASES 

This  case  of  an  irreversible  process  comes  under  group  d. 
Concerning  this  phenomenon  J.  W.  GIBBS  established  the  follow- 
ing proposition: 

"  The  entropy  of  a  mixture  of  gases  is  the  sum  of  the  entropies 
which  the  individual  gases  would  have,  if  each  at  the  same  temper- 
ature occupied  a  volume  equal  to  the  total  volume  of  the  mixture." 

That  the  total  entropy  will  be  larger  as  a  result  of  the  mixing 
detailed  under  d,  p.  73,  maybe  inferred  from  the  following  consid- 
eration: When  two  gases  are  thus  brought  together,  it  is  more 
probable  that  in  any  part  of  the  total  space  available  for  this 
mixture  there  will  be  found  both  kinds  of  molecules  than  only 
one  kind  of  these  molecules. 

But  this  irreversible  process  can  be  explained  in  a  more  dis- 
tinctly physical  way.  The  two  gases  are  originally  at  the  same 
pressure  and  temperature;  they  mix  without  other  changes 
occurring  in  surrounding  bodies;  the  mixture  (when  diffusion  is 
completed)  is  at  the  same  pressure  and  temperature  as  the 
original  gaseous  constituents.  Considering  each  gas  by  itself, 
what  has  happened  as  the  result  of  diffusion  is  that  each  gas  in 
its  final  state  occupies  a  larger  volume  than  in  its  initial,  unmixed, 
state.  The  presence  of  the  other  gas  in  the  mixture  in  no  wise 
changes  this  fact.  Of  course  this  increment  in  volume  is  accom- 
panied by  a  corresponding  decrement  in  its  pressure,  without 
change  in  temperature.  A  sort  of  isothermal  change  of  state 
has  taken  place  in  the  passage  from  one  condition  of  thermal 
equilibrium  to  the  other.  We  have  already  seen  that  then  the 
number  of  complexions  of  the  gas  increases  and  consequently 
also  its  entropy.  The  sum  of  the  increments  of  the  number  of 
complexions  separately  experienced  by  the  two  diffusing  gases 
constitutes  an  increase  in  the  total  number  of  complexions  over 
and  above  the  total  number  of  complexions  existing  in  both  gases 


78  THE  PHYSICAL   SIGNIFICANCE  OF  ENTROPY 

before  diffusion.  There  is  of  course  a  corresponding  increase  in 
entropy  due  to  such  diffusion. 

All  these  irreversible  processes  are  passages  from  less  stable 
to  more  stable  conditions,  from  less  probable  to  more  probable 
states,  or  summarizing: 

There  is  in  Nature  a  constant  tendency  to  equalize  tempera- 
ture differences,  to  convert  work  into  heat,  to  increase  disgrega- 
tion  and  to  promote  diffusion. 

This  tendency  has  also  been  described  as  the  tendency  in  Nature 
to  pass  from  concentrated  to  distributed  conditions  of  energy. 

The  four  irreversible  processes  just  discussed  are  all  sponta- 
neous ones,  i.e.,  they  occur  without  the  help  of  agencies  external 
to  the  bodies  directly  engaged  in  the  transformations. 

It  is  evident  that  the  foregoing  statements  are  really  identical, 
expressing  the  same  thought  in  different  ways. 

SECTION  C 

NEGATIVE    CHANGE    OF    ENTROPY;    SOME    OF  ITS  PHYSICAL 
FEATURES  OR  NECESSARY  ACCOMPANIMENTS 

A  negative  transformation  in  any  part  of  a  system  is  the 
diminution  of  entropy  which  it  experiences,  and  this  we  know 
means  a  diminution  in  the  number  of  complexions  of  the  part 
considered.  But  there  are  some  features  of  such  negative  trans- 
formations which,  while  they  do  not  in  themselves  constitute  any 
additional  principle,  deserve  special  mention. 

Before  we  make  such  mention,  however,  we  will  anticipate 
a  little,  and  state  the  Second  Law  in  forms  which  will  make  said 
features  obvious: 

In  an  irreversible  cycle  the  sum  of  the  changes  of  entropies 
experienced  by  all  the  bodies  concerned  is  greater  than  zero. 
When  the  cycle  is  reversible  in  all  of  its  parts,  then  said  sum 
of  entropy  changes  is  equal  to  zero. 

A  corollary  from  this  theorem  is  that,  in  a  cycle, 


AND   OF   THE  SECOND  LAW  79 

All  the  negative  transformations  present  £  all  the  positive 
transformations  that  occur. 

f  an  irreversible  1 

When  there  is  simply  \  .,  1       }•  process    without    the 

[    a  reversible    J  r 

cyclic  feature,  then  the  sum  of  the  entropies  of  all  the  bodies  par- 
ticipating in  any  one  occurrence  is,  at  the  end  of  the  change  of  con- 

\  greater  than  ] 

dition  \  j  .        \  that  at  the  beginning. 

[     equal  to      J 

From  this  we  see  that  a  negative  change  of  entropy  always 
keeps  company  with  an  equal  or  greater  positive  change  of  entropy. 

Again,  for  sake  of  simplicity,  use  a  gas  as  an  illustration; 
then  we  may  say:  (i)  Every  possible  negative  transformation  in  a 
gas  is  always  accompanied  by  a  net  positive  transformation  in 
the  other  and  necessary  external  agencies.  (2)  All  possible 
negative  transformations  in  a  gas  are  reversible  ones.  We  here 
use  the  word  possible  because  there  is  an  impossible  class  of 
negative  transformations,  namely,  those  which,  so  far  as  order 
and  directness  are  concerned,  are  the  very  opposites  of  the  so-called 
spontaneous  changes  of  state. 

It  will  suffice  here  to  enumerate  these  opposites:  Without 
external  help  (a)  to  pass  heat  from  a  cold  to  a  hot  body,  (b)  to 
decrease  the  volume  of  a  gas,  (c)  to  convert  the  heat  of  friction 
directly  back  into  the  work  which  called  it  forth,  (d)  to  separate  the 
gaseous  constituents  of  a  mixture. 

By  way  of  contrast  we  may  add,  that  the  so-called  spontaneous 
(irreversible)  processes  were  all  positive  transformations  which 
took  place  without  any  change  whatever  in  surrounding  bodies. 


80  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 


SECTION  D 

PHYSICAL  SIGNIFICANCE  OF  THE  EQUIVALENTS  FOR  GROWTH  or 
ENTROPY  GIVEN  ON  PAGES  42-43 

According  to  equivalent  (i)  growth  of  entropy  is  a  passage 
from  more  to  less  available  energy.  The  comment  already  made 
on  p.  42  indicates  sufficiently  that  this  increase  in  unavailability 
is  due  to  the  growth  of  the  ungovernable  features  of  molecular 
motions  as  number  of  complexions  increases. 

Equivalent  (2)  states  growth  of  entropy  to  be  a  passage  from 
a  concentrated  to  a  distributed  condition  of  energy.  In  this 
scattered  state  the  energy  is  certainly  less  controllable  and  for 
the  same  reason  as  that  given  concerning  equivalent  (i). 

Equivalent  (3)  is  based  on  the  idea  of  irreversibility,  and  we 
saw  on  p.  36  that  the  growth  in  the  number  of  complexions  is 
the  measure  as  well  as  the  criterion  of  irreversibility.  This  growth 
is  therefore  a  sufficient  and  necessary  feature  of  this  equivalent. 

The  equivalents  grouped  under  (4)  are  all  based  on  the  theory 
of  probabilities.  We  have  seen  on  pp.  36,  62,  and  elsewhere, 
that  the  probability  W  of  a  state  is  the  logarithm  of  the  number 
of  complexions  of  the  state.  This  number  is  therefore  a  necessary 
feature  of  this  set  of  equivalents  and  hence  constitutes  its 
physical  significance. 

The  set  of  equivalents  grouped  under  (5)  are  all  closely 
related,  their  dependence  being  more  or  less  indicated  by  the 
order  in  which  they  are  there  stated.  The  outcome  of  the  series 
is  that  growth  of  entropy  corresponds  to  an  increase  in  the 
number  of  complexions. 

The  mathematical  concept  stated  under  (6)  covers  more  than 
molecular  configurations;  it  covers  configurations  whose  elements 
are  those  of  energy  as  well,  and  has  been  successfully  applied  by 
PLANCK  in  problems  dealing  with  the  energy  of  radiation.  Every 
such  configuration  has  a  number  of  complexions. 


AND   OF   THE  SECOND  LAW  81 


SECTION  E 

PHYSICAL  SIGNIFICANCE  OF  THE  MORE  SPECIFIC  STATEMENTS 
OF  THE  SECOND  LAW  GIVEN  ON  PAGES  44-47 

In  making  here  the  contemplated  comparisons  and  interpreta- 
tions we  must  keep  in  mind  the  three  helpful  propositions  given 
on  p.  44. 

The  conservative  statement  under  (i)  is  confessedly  based 
on  the  Calculus  of  Probabilities  as  applied  to  a  mechanical  system. 
We  repeat  here  therefore  what  was  said  about  (4)  of  the  preceding 
series  of  equivalents,  namely,  that  the  number  of  complexions 
of  the  state  is  a  necessary  feature  of  this  statement  of  the  second 
law  and  therefore  constitutes  its  physical  significance. 

The  statement  under  (2)  is  a  common  one.  As  each  of  the 
exact  definitions  of  the  entropy  for  every  natural  event  has  been 
shown  to  depend  solely  on  the  number  of  complexions  of  a  system 
(all  the  bodies  participating  in  the  event  being  considered  a  part 
of  the  system)  we  have  here  likewise  in  this  number  an  adequate 
physical  explanation  of  the  second  law. 

Statements  (3),  (8)  and  (9)  have  already  been  derived  and 
explained  in  this  presentation  (see  pp.  45,  46)  as  the  result  of  the 
growth  of  the  number  of  complexions  in  every  natural  event, 
when  all  the  bodies  participating  in  the  event  are  considered. 

Statement  (4)  is  only  a  slight  variation  of  (3)  and  needs  no 
special  comment  here. 

The  same  may  be  said  of  the  three  forms  under  (5). 

The  statement  in  (6)  is  only  a  corollary  resulting  from  the 
use  of  (3)  or  (4)  or  (5). 

The  statement  in  (7)  of  the  second  law  may  be  objected  to 
because  the  underlying  definitions  are  not  entirely  free  from 
ambiguity  and  because  it  lacks  a  scientifically  general  character. 
But  it  expresses  compactly  a  matter  of  great  consequence  in 
technical  circles.  Moreover  its  explanation  in  our  physical  terms 


82  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

is  very  simple  and  direct,  viz.,  when  waste  is  incurred  there  is  a 
growth  in  the  number  of  complexions,  the  complexity  of  the 
molecular  motions  has  been  increased.  Less  of  the  stored-up 
energy  is  available,  less  is  capable  of  being  directed  into  certain 
technical  channels.  Evidently  the  greater  the  complexity  of 
the  molecular  motions  the  less  governable  they  are  by  any  direct 
external  force  or  influence  we  can  bring  to  bear.  This  is  because 
we  are  unable  to  act  directly  on  the  individual  molecules  and 
sway  them  to  our  special  technical  purpose.  Our  external  forces 
and  agencies  can  only  operate  on  the  aggregates  comprising  our 
system  and  must  obey  the  one-sided  law  imposed  on  all  such 
aggregates. 


AND  OF  THE  SECOND  LAW  83 


PART  IV 
SUMMARY: 

THE    CONNECTION    BETWEEN    PROBABILITY,   IRREVERSIBILITY, 
ENTROPY  AND  THE  SECOND  LAW 

SECTION  A 

(i)  Prerequisites  and    Conditions  Necessary  for  the  Application 
of  the  Theory  of  Probabilities 

THESE  may  be  briefly  stated  to  be  (a)  atomic  theory,  (b)  the 
likeness  of  particles  (or  elements),  (c)  very  numerous  particles, 
and  (d)  "  elementary  chaos." 

The  first  prerequisite  is  that  the  body  (here  a  gas)  is  made  up 
of  small,  discrete  particles.  This  atomic  theory  has  long  been 
the  foundation  stone  of  chemistry,  and  is  again  coming  into 
deserved  esteem  in  Physics  pure  and  simple.  (See  simple  and 
clear  article  in  Harper's  Monthly,  June,  19/0).  But  this  minute 
subdivision  must  be  accompanied  by  the  particles  being  of  the  same 
kind,  or  at  least  belonging  to  comparatively  few  groups,  each  con- 
taining many  particles  of  the  same  sort.  This  likeness  is  necessary; 
for  only  from  this  likeness  results  law  and  order  in  the  whole 
from  disorder  in  the  parts.  If  the  constituents  were  of  many 
different  kinds,  the  results  in  the  aggregate  would  not  be  so  simple 
as  we  actually  find  them  to  be.  There  is  an  example  of  this  sort 
of  complexity  in  chemistry.  We  have  already  intimated  that  the 
particles  of  each  kind  must  be  very  numerous,  but  special  emphasis 
must  be  laid  on  this  prerequisite.  If  we  ask  how  numerous  these 


84  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

elements  must  be  in  order  that  the  Theory  of  Probabilities  may 
be  applicable,  the  answer  is,  as  many  constituents  as  are  necessary 
to  determine  the  mean  values  which  define  the  state  in  the  macro- 
scopic sense  (i.e.,  in  the  aggregate  condition).  An  idea  of  the 
extent  to  which  Nature  carries  this  subdivision  is  furnished  by 
the  fact  that  one  grain  (avordupois)  of  air,  under  standard  con- 
ditions, contains, 

14  (io20)  molecules, 

•-, 
i.e.,  millions  of  billions  of  particles! 

The  last  one  of  said  prerequisites  is  "  elementary  chaos/ r 
and  needs  further  elucidation  and  limitation;  we  will  therefore 
go  into  this  feature  at  greater  length. 

BOLTZMANN  has  used  the  term  "  molekular-ungeordnet " 
(molecularly  disordered)  to  designate  this  chaotic  condition  of 
the  particles,  and  PLANCK  has  introduced  a  more  general  term 
still,  "  elementar-ungeordnet "  (elementary  disorder  or  chaos) 
in  order  to  make  the  method  applicable  to  phenomena  like  radia- 
tion, in  which  the  elements  are  not  atoms  or  particles  but  partial 
oscillations  of  different  periods.  The  essence  of  the  matter  seems 
to  consist  in  excluding  from  consideration  all  such  regularities 
in  the  conditions  of  the  elements  as  would  lead  to  results  at  variance 
with  the  well-known  laws  of  physical  phenomena,  justifying  this 
exclusion  by  the  assumption  that  no  such  elementary  regularities 
obtain  in  Nature.  This  only  means  that  not  all  of  the  many 
molecular  arrangements,  which  are  conceivable  from  the  purely 
mechanical  standpoint,  are  actually  realized.  For  instance, 
in  an  isolated  gaseous  system  we  could  conceive  of  a  succession 
of  elementary  states  at  variance  with  the  principle  of  conservation 
of  energy;  such  a  set  would  obviously  not  be  realized.  This 
exclusion  or  limitation  leaves  room  for  various  hypotheses  as  to 
said  elementary  disarrangement,  but  to  be  admissible  they  must 
all  permit  of  the  legitimate  application  of  the  Theory  of  Prob- 
abilities, the  best  one  being  ultimately  determined  by  its  agreement 


AND  OF  THE  SECOND  LAW  85 

in  the  whole  with  known  facts  or  laws.  Evidently  by  prearrange- 
ment  and  precomputation  there  could  be  obtained  molecular 
arrangements  which  would  establish  long-continued  regularities, 
which  would  furnish  mean  results  in  the  aggregate,  that  would 
be  at  variance  with  the  well-known  behavior  of  Nature.  All 
such  cases  are  here  excluded. 

According  to  PLANCK  the  unregulated,  confused  and  whirring 
intermingling  of  very  many  atoms  (in  the  case  of  a  monatomic 
gas)  is  the  prerequisite  for  the  validity  of  this  hypothesis  of 
"  elementary  chaos." 

(2)  Differences    in  the    States    of  "Elementary    Chaos" 

When  we  consider  the  general  state  of  a  gas  we  need  not  think 
of  the  state  of  equilibrium,  for  this  is  still  further  characterized 
by  the  condition  that  its  entropy  is  a  maximum.1 

Hence  in  the  general  or  unsettled  state  of  the  gas  an  unequal 
distribution  of  density  may  prevail,  any  number  of  arbitrarily 
different  streams  (whirls  and  eddies)  may  be  present,  and  we 
may  in  particular  assume  that  there  has  taken  place  no  sort  of 
equalization  between  the  different  velocities  of  the  molecules. 
To  conceive  of  said  differences  we  may  assume  beforehand,  in 
perfectly  arbitrary  fashion,  the  velocities  of  the  molecules  as  well 
as  their  co-ordinates  of  location.  But  there  must  exist  (in  order 
that  we  may  know  the  state  in  the  macroscopic  sense),  certain 
mean  values  of  density  and  velocity,  for  it  is  through  these  very 
mean  values  that  the  state  is  characterized  from  the  aggregate 
(macroscopic)  standpoint.  The  differences  that  do  exist  in  the 
successive  stages  of  disorder  of  the  unsettled  state  are  mainly 
due  to  the  molecular  collisions  that  are  constantly  taking  place, 
thus  changing  the  velocity  and  locus  of  each  molecule. 

Let  us  for  sake  of  brevity  speak  of  the  state  of  permanence 

1  The  rest  of  the  paragraph  is  a  repetition  of  what  was  stated  at  middle  of  p.  19. 


86  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

finally  attained  by  this  chaotic  mass  as  the  normal  state,  and  all 
the  preceding  chaotic  states  as  abnormal  states. 

(3)  Number  of  complexions,  or  probability,  of  a  chaotic  state. 

It  was  shown,  in  an  earlier  portion  of  this  presentation,  that 
each  such  chaotic  state  (abnormal  or  normal)  is  characterized 
by  its  number  of  complexions,  which  is  determined  by  the  Theory 
of  Probabilities.  This  number  is  a  variable  one  for  the  successive 
abnormal  states  and  is  a  fixed  and  a  maximum  one  (under  given 
external  conditions)  for  the  normal  state.  Now  BOLTZMANN 
(by  the  application  of  the  Theory  of  Probabilities  to  this  chaotic 
state)  has  shown  that  the  means  of  these  states  vary  in  one  direction 
only,  in  such  a  way  that  the  probable  number  of  complexions 
of  the  successive  abnormal  states  continually  grows  till  it  attains 
its  maximum  in  the  normal  and  permanent  state. 

SECTION  B 
IRREVERSIBILITY 

This  one-sidedness  of  the  average  action  or  flux  constitutes  and 
sharply  defines  what  is  meant  by  irreversibility.  It  does  not 
imply  that  the  motion  of  any  particular  atom  cannot  be  reversed, 
but  that  the  order  in  which  these  averages  (or  the  number  of 
complexions)  occur  cannot  be  reversed.  We  have  here  a  process, 
consisting  of  a  number  of  separately  reversible  processes,  which 
proves  to  be  irreversible  in  the  aggregate.  This  is  not  the  only 
possible  characterization  of  the  property  of  irreversibility  inherent 
in  all  natural  events,  but  is  perhaps  as  general  and  exact  a  one 
as  can  be  enunciated.  Superficially  speaking,  from  the  confused 
and  irregular  motions  contemplated,  it  is  quite  evident  that  this 
succession  of  whirls  and  eddies  cannot  be  worked  directly  back- 
ward to  bring  about,  in  reverse  order,  the  finite  physical  state 
which  initiated  them;  for  the  effecting  of  such  an  opposite  change 


AND  OF  THE  SECOND  LAW  87 

would  demand  a  co-operation  and  concert  of  action  on  the  part  of 
the  elementary  constituents  which  is  felt  to  be  quite  impossible. 
It  will  not  be  so  general  and  scientific,  but  perhaps  more  easily 
apprehended,  if  we  put  this  result  in  terms  of  human  effort, 
namely,  "  by  asserting  that  any  process  is  irreversible  we  assert 
that  by  no  means  within  our  present  or  future  power  can  we 
reverse  it,  i.e.,  we  cannot  control  the  individual  molecules." 


SECTION  C 
ENTROPY 

We  have  seen  above  that  the  inevitable  growth  in  the  number 
of  complexions  is  the  mark  of  irreversibility;  the  number  of 
complexions  at  any  stage  can  also  in  a  certain  sense  be  regarded 
as  the  measure,  index  or  determinant  of  that  stage  or  state  of  the 
system  of  elements  under  consideration.  Any  function  of  the 
number  of  complexions  can  be  regarded  as  such  measure,  index 
or  determinant.  Now  it  has  been  shown  by  BOLTZMANN  that 
the  expression  found  thermodynamically  for  the  quantity  called 
entropy  differs  only  by  a  physically  insignificant  constant  from 
the  logarithm  of  said  number  of  complexions.  But  the  latter 
may  properly  be  regarded  as  a  true  measure  of  the  probability 
of  the  system  being  in  the  state  considered.  BOLTZMANN  has 
defined  the  entropy  of  a  physical  system  as  the  logarithm  of  the 
probability  of  the  mechanical  condition  of  the  system  and  PLANCK 
has  cast  it  into  the  numerical  form, 

S=!i.35  loge("probability")io-16-f  constant  K 
=  1.35  loge  (number  of  complexions)  io-16  + const.  K; 

where  S  is  the  entropy  of  any  natural  state  of  the  body  and  K  is 
an  arbitrary  constant,  the  numerical  value  of  the  first  term  of  the 
second  member  is  the  quotient  of  the  energy  (expressed  in  ergs) 


88  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

divided  by  the  temperature  (in  centigrade  degrees).  In  English 
units  and  the  F.P.S.  system  this  numerical  value  is  5.50  (io~24). 
From  the  whole  development  we  see  that  entropy  S  depends 
only  on  the  number  of  complexions;  it  should  not  be  considered, 
as  is  sometimes  done,  as  of  the  same  dimensions  of  energy  or 
anything  that  may  generally  be  called  a  factor  of  energy. 

SECTION  D 
THE  SECOND  LAW 

It  is  evident  that  all  these  results  have  for  their  original  basis 
the  Theory  of  Probabilities.  Consequently,  because  these  con- 
clusions are  thus  based,  they  must  be  interpreted  according  to  the 
general  method  underlying  this  Theory.  This  method  essentially 
is  the  determination  of  average  (mean)  values  and  calling  them 
the  probable  ones.  We  therefore  conclude  that  each  state  is 
characterized  by  the  mean  number  of  complexions  belonging 
to  that  state,  that  is,  by  this  mean  number  which  changes  always 
in  a  one-sided  way,  ever  in  the  same  sense,  inasmuch  as  it  inevitably 
and  invariably  grows  till  the  normal,  settled  condition  is  reached. 

For  the  sake  of  clarity  we  must  keep  in  mind  that  the  motions 
of  the  individual  atoms  are  reversible  and  that  in  this  sense  the 
irreversible  processes  are  reduced  to  reversible  ones.  But  the 
process  as  a  whole  is  not  reversible  because,  by  the  very  act  of 
complete  reversal,  we  would  suspend  the  general,  chaotic  character 
of  the  elementary  motions  and  give  them  to  this  extent  a  special, 
prearranged  feature  which  would  be  more  or  less  hostile  to  the 
original  definition  of  "  elementary  chaos."  The  irreversibility 
is  not  in  the  elementary  events  themselves,  but  solely  in  their 
irregular  arrangement.  It  is  this  which  guarantees  the  one-sided 
change  of  the  mean  value  characteristic  of  each  one  of  the  successive 
states  of  the  process. 

Now  remembering  that  the  kernel  of  the  Second  Law  is  that 
all  processes  in  Nature  are  irreversible,  or,  that  all  changes  in 


.     AND  OF  THE  SECOND  LAW  89 

Nature  vary  in  one  direction  only,  we  can,  in  the  light  of  what  of 
has  just  preceded,  repeat  the  following  precise,  scientific  state- 
ment: 

"  The  Second  Law,  in  its  objective-physical  form  (freed  from 
all  anthropomorphism)  refers  to  certain  mean  values  which  are 
found  from  a  great  number  of  like  and  '  chaotic  '  elements." 

If  we  now  go  back  to  what  constitutes  the  kernel  of  the  Second 
Law,  we  will  see  the  relevance  and  force  of  PLANCK'S  enunciation 
of  this  law: 

"It  is  not  possible  to  construct  a  periodically  functioning 
motor  which  effects  nothing  more  than  the  lifting  of  a  load  and 
the  cooling  of  a  heat  reservoir." 

The  proof  of  this  is  purely  experimental  and  cumulative,  and 
the  same  may  be  said  of  the  earlier  statement  of  this  law,  "  all 
changes  in  Nature  vary  in  one  direction  only."  The  character 
of  this  proof  is,  moreover,  exactly  like  that  for  the  First  Law, 
the  Conservation  of  Energy,  and  has  the  same  sort  of  validity. 

When  we  compared  and  interpreted  the  current  statements  of 
the  Second  Law  (pp.  44-47)  we  enunciated  and  made  use  of 
three  helpful  propositions  that  will  now  be  repeated : 

(a)  All  cases  of  irreversibility  stand  or  fall  together;  if  any  one 
can  be  reversed  all  can  be  reversed. 

(b)  Any  general  consequence  of  any  one  correct  statement  of 
the  Second  Law  may  be  regarded  as  itself  a  valid  and  com- 
plete statement  of  the  Second  Law. 

(c)  The  summary  of  all  the  necessary  prerequisites  (or  con- 
ditions) for  determining  Entropy  may  be  regarded  as  a 
complete  and  valid  statement  of  the  Second  Law. 

In  this  connection  it  will  also  be  helpful  to  remember  PLANCK'S 
statement:  "  In  order  that  a  process  may  be  truly  reversible  it 
will  not  suffice  to  declare  that  the  mediating  body  is  directly 
reversible,  but  that  at  the  end,  everywhere  in  the  whole  of  Nature, 
the  same  state  must  be  restored  which  existed  at  the  beginning 
of  said  reversible  process." 


90  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

As  regards  the  use  of  helpful  proposition  (a) : 

We  know  that  PLANCK'S  motor  statement  of  the  Second  Law 
was  grounded  on  the  well-known  irreversible  passage  of  heat  from 
a  cold  to  a  hot  body.  But  to  show  the  mutual  interdependence  (a) 
of  one  irreversible  change  on  every  other,  we  will  instance  in  illus- 
tration the  case  of  a  frictional  event,  or  the  conversion  of  mechan- 
ical work  into  heat. 

If  this  frictional  occurrence  could  by  any  simple  or  complex 
apparatus  be  made  completely  reversible  so  that  everywhere,  in 
the  whole  of  Nature,  the  same  state  would  be  restored  which 
existed  at  the  beginning  of  the  frictional  occurrence,  then  such 
an  apparatus  would  be  the  motor  contemplated  in  PLANCK'S 
statement  of  the  Second  Law,  for  this  periodically  running 
apparatus  would  convert  heat  into  work  without  any  other  change 
remaining.  A  similar  line  of  argument,  with  a  similar  result, 
could  be  pursued  with  every  other  case  of  irreversibility  that  could 
be  adduced.  It  is  evident  that,  with  the  help  of  the  above-given 
propositions  (a),  (6),  and  (c),  the  Second  Law  can  be  cast  into 
many  other  valid  forms. 

We  close  this  presentation  of  the  meaning  of  the  Second  Law 
by  the  remark  that  this  law  has  no  independent  significance,  for 
its  roots  go  down  deep  into  the  Theory  of  Probabilities.  It  is 
therefore  conceivable  that  it  is  applicable  to  some  purely  human 
and  animate  events  as  well  as  to  inanimate,  natural  events; 
provided,  of  course,  that  the  former  possess  numerous  like  and 
uncontrolled  constituents  which  may  be  properly  characterized 
as  "  elementar-ungeordnet,"  in  other  words,  provided  the  variable 
elements  present  constitute  adequate  haphazard  for  the  Calculus 
of  Probabilities. 


AND   OF  THE  SECOND  LAW  91 


PART  V 
REACH  AND  SCOPE  OF  SECOND  LAW 

SECTION  A 

ITS  EXTENSION  TO  ALL  BODIES 

CLAUSIUS  extended  the  operation  of  the  Second  Law  or,  what 
is  the  same  thing,  the  scope  of  entropy,  to  alj.  bodies.  See  RUHL- 
MANN'S  "  Handbuch  d.  mech.  Warmtheorie,"  Vol.  I,  pp.  395-405. 

BOLTZMANN  says  in  this  connection:  "  As  regards  entropy, 
solid  and  liquid  bodies  do  not  differ  qualitatively  from  perfect 
gases;  the  discussion  of  the  entropy  of.  the  former,  however, 
presents  greater  mathematical  difficulties." 

Certain  features  of  the  entropy  of  solid  and  liquid  bodies 
have,  however,  been  derived  with  the  help  of  ideal  gases  as  tem- 
porary auxiliaries.  We  consider  this  argument  the  simplest  and 
therefore  now  give  an  outline  of  PLANCK'S  presentation  of  the 
matter.1 

PLANCK'S  PROOF  THAT  ALSO  FOR  ANY  OTHER  BODIES  THAN  GASES 
THERE  REALLY  EXISTS  A  FUNCTION  WHICH  POSSESSES  THE  CHAR- 
ACTERISTICS OF  ENTROPY;  THE  MAIN  STEPS  ARE  NUMBERED 

(i)  Expression  of  entropy  for  an  ideal  gas  and  properties  of 
entropy. 

(2)  (a)  S-M(C.logr+logv+const);    .     .     (34) 


d'Q 


(35) 


Thermodynamik,  2d  Ed.,  pp.  87-100. 


92  THE  PHYSICAL  SIGNIFICANCE   OF  ENTROPY 


where  the  elastic  forces  do  a  work  =  />dF;  strictly  speaking,  d'Q  is 
not  differential  of  Q  the  heat  supply. 

(3)  Two  gases  (i)  and  (2)  thermally  connected,  are  maintained 
at  same  temperature  but  different  pressure  and  change  adiabat- 
ically  while  experiencing  change  of  volumes;  then  it  can  be  shown 
that  for  this  finite  change,  5i+52  =  constant,  that  is  for  the  two 
gases  the   sum  of  the   final    entropies  =  Sil+S21=  Si  +$2  =  sum 
of  their  initial   entropies.    No  other  change  is  effected  in  any 
other  bodies  but  in  these  two  gases;   here  emphasis  is  laid  on 
preposition  in;   for  the  work  done  may  be  the  lifting  or  lowering 
of  a  load  and  such  change  of  location  in  rigid  bodies  involves 
no  change  of  inner  energy.    Changes  of  density  in  external  bodies 
can  be  also  avoided  by  having  the  two  gas  tanks  located  hi  a 
vacuum. 

(4)  A  similar  proposition  can  be  established  for  a  system  of  any 
number  of  gases  by  successively  treating  the  gases  in  pairs  as  above. 
The  theorem  then  reads:  "If  the  gas  system  as  a  whole  possesses 
the  same  entropy  in  two  different  states  then  the  system  can  be 
brought  from  one  state  to  the  other  in  a  reversible  manner  without 
changes  .remaining  in  other  bodies." 

(5)  We  know  that  the  expansion  of  an  ideal  gas  without  doing 
external  work  and  receiving  any  heat  supply  is  an  irreversible 
process.    The  consequence  is  that  the  entropy  of  this  gas  increases. 
It  follows  at  once  that  "  it  is  impossible  to  diminish  the  entsopy 
of  an  Meal  gas  without  changes  remaining  in  other  bodies." 

(6)  The  same  result  obtains  for  a  system  of  any  number  of 
ideal  gases.     Consequently  "there  exists  in  the  whole  of  Nature 
no  means  (be  they  of  the  mechanical,  thermal,  chemical  or  electrical 
sort)  of  diminishing  the  entropy  of  a  system  of  ideal  gases,  without 
changes  remaining  in  other  bodies." 

(7)  "If  a  system  of  ideal  gases  has  changed  to  another  state  (pos- 
sibly in  an  entirely  unknown  way)  without  changes  remaining 
in  other  bodies,  then  the  final  entropy  can  certainly  not  be  smaller, 
it  can  only  be  greater  than  or  equal  to  the  initial  condition.     In 


AND  OF  THE  SECOND  LAW  93 

the  former  case  this  process  is  an  irreversible  one,  in  the  latter 
case  a  reversible  one. 

"  Equality  of  entropy  in  the  two  states  therefore  constitutes 
a  sufficient  and  at  the  same  time  a  necessary  condition  for  the 
complete  reversibility  of  the  passage  from  one  state  to  the  other, 
provided  no  changes  are  to  remain  behind  in  other  bodies." 

(8)  "This  proposition  has  a  very  considerable  range  of  validity; 
for  there  was  expressly  no  limiting  assumption  made  concerning 
the  way  in  which  the  gas  system  reached  its  final  condition;   the 
proposition  is  therefore  valid  not  only  for  slowly  and  simply 
changing  processes  but  also  for  any  physical  and  chemical  proc- 
esses provided  at  the  end  no  changes  remained  in  any  body  out- 
side of  the  gas  system.    Nor  need  we  believe  that  entropy  of  a 
gas  has  significance  only  for  states  of  equilibrium,  provided  we  can 
suppose  the  gas  mass  (moving  in  any  way)  to  consist  of  sufficiently 
small  parts  each  so  homogeneous  that  it  possesses  entropy."  l 

Then  the  summation  must  extend  over  all  these  gas  parts. 
"  The  velocity  has  no  influence  on  the  entropy,  just  as  little  as  the 
height  of  the  heavy  gas  parts  above  a  particular  horizontal  plane." 

(9)  "The  laws  thus  far  deduced  for  ideal  gases  can  in  the 
same  way  be  transferred  to  any  other  bodies,  the  main  difference 
in  general  being  that  the  expression  for  the  entropy  of  any  body 
cannot  be  written  in  finite  magnitudes  because  the  equation  of 
condition  is  not  generally  known.    But  it  can  always  be  shown — 
and  this  is  the  decisive  point — that  for  any  other  body  there 
really  exists  a  function  possessing  the  characteristic  properties 
of  entropy." 

Now  let  us  assume  any  physically  "  homogeneous  body,  by 
which  is  meant  that  the  smallest  visible  space  parts  of  the  system 
are  completely  alike.  Here  it  does  not  matter  whether  or  no 
the  substance  is  chemically  homogeneous,  i.e.,  whether  it  consists 

1  If  the  motion  of  the  gas  is  so  turbulent  that  temperature  and  density  cannot 
be  defined,  then  we  must  have  recourse  to  BOLTZMANN'S  broader  definition  of 
entropy. 


94  THE  PHYSICAL  SIGNIFICANCE   OF   ENTROPY 

of  entirely  like  molecules,  and  consequently  it  also  does  not  here 
matter  whether  in  the  course  of  the  prospective  changes  of  state 
it  experiences  chemical  transformation.  .  .  .  When  the  substance 
is  stationary  the  whole  energy  of  the  system  will  consist  of  the 
so-called  '  inner  '  energy  Z7,  which  depends  only  on  the  mass  and 
inner  constitution  of  the  substance,  which  constitution  is  conditioned 
by  the  temperature  and  density." 

(10)  Let  us  suppose  that  with  such  a  homogeneous  body  there 
is  conducted  a  certain  reversible  or  irreversible  cycle  process  which 
therefore  brings  the  body  exactly  back  again  to  its  initial  con- 
dition. Let  the  external  influences  on  the  body  consist  in  the 
performance  of  work  and  in  heat  supply  or  withdrawal,  which 
heat  exchange  is  to  be  effected  by  any  number  of  suitable  heat 
reservoirs.  At  the  end  of  the  process  no  changes  remain  in  the 
body  itself,  only  the  heat  reservoirs  have  altered  their  state.  Now 
let  us  suppose  the  heat  carriers  in  the  reservoirs  to  be  composed 
of  purely  ideal  gases,  which  may  be  kept  at  constant  volume  or 
under  constant  pressure,  at  any  rate  only  be  subject  to  reversible 
changes  of  volume.  According  to  the  last  proved  proposition,  the 
sum  of  the  entropies  of  all  the  gases  cannot  have  become  smaller, 
for  at  the  end  of  the  process  no  changes  remain  in  any  other  body, 
not  even  in  the  body  which  completed  the  cycle  process. 

(n)  "Letd'Q  be  the  heat  gained  by  the  body  from  some  reser- 
voir in  an  element  of  time  and  T  the  temperature  of  the  reser- 
voir1 at  the  same  moment,  then  the  change  of  entropy  experienced 
by  the  reservoir  at  this  instant  will  be 


and  in  the  whole  course  of  time  all  the  reservoirs  together  will 
experience  the  entropy  change 


1  It  does  not  here  matter  what  the  temperature  of  the  body  is  at  this  instant. 


AND  OF   THE  SECOND  LAW  95 

and  then  we  know  that  there  must  be  satisfied  the  condition 


_S>0     or      S<0,    ____    (36) 

which  is  the  form  in  which  CLAUSIUS  first  enunciated  the  Second 
Law. 

(12)  Another  condition  for  the  process  considered  is  furnished 
by  the  First  Law.  For  each  element  of  time  d'Q+A  =  dU,  where 
U  is  the  inner  energy  of  the  body  and  A  the  work  expended  upon 
it  in  an  element  of  time  by  external  means. 

Now  let  us  consider  a  more  special  case  in  which  the  external 
pressure  at  each  instant  is  equal  to  the  pressure  p  of  the  suppos- 
edly stationary  body.  Then  the  external  work  will  be  represented 

by 

.......    (37) 


and  then  it  follows  that 

.     .....     (38) 


(13)  Furthermore  let  the  temperature  of  each  heat  reservoir, 
at  the  instant  when  it  comes  into  use,  be  equal  to  the  simultaneous 
temperature  of  the  body;  then  the  cycle  process  becomes  a  revers- 
ible one  and  the  inequality  of  the  second  law  becomes  an  equality, 


(39) 


and  substitution  of  above  value  for  d'Q  gives 


^dU  +  pdV 

--  —  =o  .......     (40) 


In  this  equation  there  occur  only  quantities  referring  to  the 
state  of  the  body  itself  and  therefore  it  can  be  interpreted  without 


96  THE  PHYSICAL  SIGNIFICANCE  OF  ENTROPY 

any  reference  to  the  heat  reservoirs.     It  contains  the  following 
proposition : 

"  If  a  homogeneous  body  by  suitable  treatment  is  allowed  to 
pass  through  a  series  of  continuous  states  of  equilibrium  and 
thus  finally  to  come  back  to  its  initial  condition,  the  summation 
of  the  differential, 

dU+pdV 


for  all  the  changes  of  state  will  be  equal  to  zero.  From  this  follows 
at  once  that  if  the  change  of  state  is  not  allowed  to  continue  to 
the  restoration  of  the  initial  condition  (i),  but  is  stopped  at  any 
state  (2),  the  value  of  the  sum 

'*dU+pdV 

— TJ£ — »     (41) 


will  depend  solely  on  the  final  state  (2)  and  on  the  initial  state 
(i),  and  not  on  the  course  of  the  passage  from  i  to  2."  l 

"  The  last  expression  is  called  by  CLAUSIUS  the  entropy  of 
the  body  in  state  2,  referred  to  state  i  as  the  zero  state.  The 
entropy  of  a  body  in  a  particular  state  is,  therefore,  like  energy, 
completely  determined  down  to  an  additive  constant  depending 
on  the  choice  of  the  zero  state." 

(14)  "  Let  us  again  designate  the  entropy  by  S,  then 


^dU  +  pdV 

o  — 


1  This  is  evident  from  the  fact  that  the  quantities  U,  p,  V,  and  T,  under  the 
integral  are  each  a  function  of  the  state  only  and  do  not  depend  on  its  past  history. 
This  falls  far  short  of  being  true  for  turbulent  states,  for  which  it  is  difficult  to 
get  p  and  T.  PLANCK  does  not  make  the  preceding  statement,  but  gives  instead 
a  rigorous  proof  based  on  cyclical  considerations. 


AND  OF    THE  SECOND   LAW  97 

or,  what  amounts  to  the  same  thing,  by 

,„    dU+pdV 


(42) 


which  reduced  to  the  unit  of  mass  becomes 

du  +  pdv 


(43) 


"  This  is  evidently  identical  with  the  value  found  for  an  ideal 
gas.  But  it  is  equally  applicable  to  every  body  when  its  energy 
U  =  Mu  and  volume  V=Mv  are  known  as  functions,  say,  of  p 
and  T,  for  the  expression  for  entropy  can  then  be  directly  deter- 
mined by  integration.  But  since  these  functions  are  not  com- 
pletely known  for  any  other  substance  we  must  in  general  rest 
content  with  the  differential  equation.  For  the  present  proof, 
however,  and  for  many  applications  of  the  Second  Law  it  suffices 
to  know  that  this  differential  equation  really  contains  a  unique 
definition  of  entropy." 

As  with  an  ideal  gas,  we  can  now  always  speak  of  the  entropy 
of  any  substance  as  a  certain  finite  magnitude  determined  by 
the  values  of  the  temperature  and  volume  at  the  instant,  and 
can  so  speak  even  when  the  substance  experiences  any  reversible 
or  irreversible  change.  Moreover,  the  differential  equation  (43) 
is  applicable  to  any  change  of  state,  even  an  irreversible  one. 

In  thus  applying  the  idea  of  entropy  there  is  no  conflict  with 
its  derivation.  The  entropy  of  a  state  is  measured  by  a  reversible 
process  which  conducts  the  body  from  its  present  state  to  the 
zero  state,  but  this  ideal  process  has  nothing  to  do  with  the  changes 
of  state  that  the  body  has  experienced  or  is  going  to  experience." 

"  On  the  other  hand,  we  must  emphasize  that  differential 
equation  (43)  for  ds  is  valid  only  for  changes  of  temperature  and 
volume  and  is  not  so  for  changes  of  mass  or  of  chemical  composi- 


98  THE  PHYSICAL   SIGNIFICANCE    OF  ENTROPY 

tion.     For   changes  of  the   latter  sort  were  never  considered  in 
denning  entropy." 

(15)  "  Finally,  we  may  designate  the  sum  of  the  entropies  of 
several  bodies  as  the  entropy  of  the  system  of  all  the  bodies, 
provided  the  system  can  be  subdivided  into  infinitesimal  elements 
for  which  uniform  density  and  temperature  can  be  assumed; 
but  velocity  and  force  of  gravitation  do  not  at  all  enter  into  the 
expression  for  entropy." 


SECTION  B 
GENERAL  CONCLUSION  AS  TO  ENTROPY  CHANGES 

Now  that  the  existence  and  value  of  entropy  have  been  estab- 
lished for  every  state  of  any  body,  we  can  proceed  to  draw  general 
conclusions  in  much  the  same  way  as  that  followed  with  the  ideal 
gases.  The  general  result  is : 

"IT  IS  IN  NO  WAY  POSSIBLE  TO  DIMINISH  THE  ENTROPY  OF  A 
SYSTEM  OF  BODIES  WITHOUT  HAVING  CHANGES  REMAIN  IN  OTHER 
BODIES." 

If  we  admit  to  the  system  all  the  bodies  participating  in  the 
process,  this  theorem  becomes: 

"  Every  physical  and  chemical  process  occurring  in  Nature 
takes  place  in  such  a  way  that  there  is  an  increase  in  the  sum  of 
the  entropies  of  the  bodies  in  any  way  participating  in  the  process" 

We  will  close  with  BOLTZMANN'S  statement: 

"  The  driving  motive  (or  impelling  cause)  in  all  natural  events 
is  the  difference  between  the  existing  entropy  and  its  maximum 
value" 


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-  Workshop  Wrinkles  ."  

.  .  Svo, 

*I    00 

Bruce.  E.  M.     Pure  Food  Tests.  . 

i2mo, 

*i   25 

D.   VAN   NOSTRANDy  COMPANY'S   SHORT  TITLE   CATALOG        5 

Bruhns,  Dr.     New  Manual  of  Logarithms 8vo,  half  morocco,  2  50 

Brunner,   R.     Manufacture   of   Lubricants,   Shoe   Polishes  and  Leather 

Dressings.     Trans,  by  C.  Salter 8vo,  *3  oo 

Bulman,  H.  F.,  and  Redmayne,  R.  S.  A.     Colliery  Working  and  Manage- 
ment  8vo,  6  oo 

Burgh,  N.  P.     Modern  Marine  Engineering 4to,  half  morocco,  10  oo 

Burt,  W.  A.     Key  to  the  Solar  Compass i6mo,  leather,  2  50 

Burion,  F.  G.     Engineering  Estimates  and  Cost  Accounts i2mo,  *i  50 

Buskett,  E.  W.     Fire  Assaying i2mo,  *i  25 

Cain,  W.     Brief  Course  in  the  Calculus i2mo,  *i  75 

—  Theory  of  Steel-concrete  Arches  and  of  Vaulted  Structures.     (Science 

Series.) i6mo,  o  50 

Campin,  F.     The  Construction  of  Iron  Roofs 8vo,  2  oo 

Carpenter,  R.  C.,  and  Diederichs,  H.     Internal  Combustion  Engines. 8vo,  *5  oo 

Carter,  E.  T.     Motive  Power  and  Gearing  for  Electrical  Machinery  .  .  8vo,  *5  oo 

Carter,  H.  A.     Ramie  (Rhea),  China  Grass i2mo,  *2  oo 

Carter,  H.  R.     Modern  Flax,  Hemp,  and  Jute  Spinning 8vo,  *3  oo 

Cathcart,  W.  L.     Machine  Design.     Part  I.  Fastenings 8vo,  *3  oo 

Cathcart,  W.  L.,  and  Chaffee,  J.  I.     Course  of  Graphic  Statics 8vo  (In  Press.) 

Caven,  R.  M.,  and  Lander,  G.  D.     Systematic  Inorganic  Chemistry.  i2mo,  *2  oo 

Chambers'  Mathematical  Tables. ., 8vo,  i  75 

Charnock,  G.  F.     Workshop  Practice.     (Westminster  Series.).  .  .  .8vo  (In  Press.) 

Charpentier,  P.     Timber 8vo,  *6  oo 

Chatley,  H.     Principles  and  Designs  of  Aeroplanes.    (Science    Series.) 

i6mo,  o  50 

-  How  to  Use  Water  Power i2mo,  *i  oo 

Child,  C.  T.     The  How  and  Why  of  Electricity i2mo,  i  oo 

Christie,  W.  W.     Boiler- waters,  Scale,  Corrosion,  Foaming 8vo,  *3  oo 

—  Chimney  Design  and  Theory 8vo,  *3  oo 

—  Furnace  Draft.     (Science  Series.) i6mo,  o  50 

Church's  Laboratory  Guide.     Rewritten  by  Edward  Kinch* 8vo,  *2  50 

Clapperton,  G.     Practical  Papermaking 8vo,  2  50 

Clark,  C.  H.     Marine  Gas  Engines. (In  Press.) 

Clark,  D.  K.     Rules,  Tables  and  Data  for  Mechanical  Engineers 8vo,  5  oo 

—  Fuel:  Its  Combustion  and  Economy i2mo,  i  50 

—  The  Mechanical  Engineer's  Pocketbook i6mo,  2  oo 

-  Tramways:  Their  Construction  and  Working 8vo,  7  50 

Clark-  J.  M.     New  System  of  Laying  Out  Railway  Turnouts i2mo,  i  oo 

Clausen-Thue,  W.     ABC  Telegraphic  Code.     Fourth  Edition i2mo,  *5  oo 

Fifth  Edition 8vo,  *7  oo 

—  The  A  i  Telegraphic  Code 8vo,  *7  50 

Cleemann,  T.  M.     The  Railroad  Engineer's  Practice i2mo,  *i  50 

Clevenger,  S.  R.     Treatise    on   the    Method    of    Government    Surveying. 

i6mo,  m.orocco 2  50 

Clouth,  F.  Rubber,  Gutta-Percha,  and  Balata 8vo,  *s  oo 

Coffin,  J.  H.  C.  Navigation  and  Nautical  Astronomy i2mo,  *3  50 

Cole,  R.  S.  Treatise  on  Photographic  Optics i2mo,  2  50 

Coles- Finch,  W.  Water,  Its  Origin  and  Use .' .  8vo,  *5  oo 

Collins,  J.  E.  Useful  Alloys  and  Memoranda  for  Goldsmiths,  Jewelers. 

i6mo o  50 


6        D.   VAN    NOSTRAND    COMPANY'S   SHORT  TITLE  CATALOG 

Constantine,  E.     Marine  Engineers,  Their  Qualifications  and  Duties. .  8vo,  *2  oo 

Cooper,  W.  R.     Primary  Batteries 8vo,  *4  oo 

—  "  The  Electrician  "  Primers 8vo,  *5  oo 

Copperthwaite,  W.  C.     Tunnel  Shields 4to,  *o  oo 

Corey,  H.  T.     Water  Supply  Engineering 8vo  (In  Prc**.) 

Cornwall,  H.  B.     Manual  of  Blow-pipe  Analysis 8vo,  *2  50 

Cowell,  W.  B.     Pure  Air,  Ozone,  and  Water i2mo,  *2  oo 

Crocker,  F.  B.     Electric  Lighting.     Two  Volumes.     8vo. 

Vol.    I.     The  Generating  Plant 3  oo 

Vol.  II.     Distributing  Systems  and  Lamps 3  oo 

Crocker,  F.  B.,  and  Arendt,  M.     Electric  Motors 8vo,  *2  50 

Crocker,  F.  B.,  and  Wheeler,  S.  S.     The  Management  of  Electrical  Ma- 
chinery  i2mo,  *i  oo 

Cross,  C.  F.,  Be  van,  E.  J.,  and  Sindall,  R.  W.     Wood  Pulp  and  Its  Applica- 
tions.    (Westminster  Series.) 8vo  (In  Press.) 

Crosskey,  L.  R.     Elementary  Perspective ' 8vo,  i  oo 

Crosskey,  L.  R.,  and  Thaw,  J.     Advanced  Perspective 8vo,  i   50 

Davenport,  C.     The  Book.     (Westminster  Series.) 8vo,  *2  oo 

Davies,  E.  H.     Machinery  for  Metalliferous  Mines 8vo,  8  oo 

Davies,  D.  C.     Metalliferous  Minerals  and  Mining 8vo,  5  oo 

Davies,  F.  H.     Electric  Power  and  Traction 8vo,  *2  oo 

Dawson,  P.     Electric  Traction  on  Railways 8vo,  *Q  oo 

Day,  C.     The  Indicator  and  Its  Diagrams. : i2mo,  *2  oo 

Deerr,  N.     Sugar  and  the  Sugar  Cane 8vo,  *3  oo 

Deite,  C.     Manual  of  Soapmaking.     Trans,  by  S.  T.  King 4to,  *5  oo 

De  la  Coux,  H.   The  Industrial  Uses  of  Water.   Trans,  by  A.  Morris.  .8vo,  *4  50 

Del  Mar,  W.  A.     Electric  Power  Conductors 8vo,  *2  oo 

Denny,  G.  A.     Deep-level  Mines  of  the  Rand 4to,  *io  oo 

-  Diamond  Drilling  for  Gold , *5  oo 

Derr,  W.  L.     Block  Signal  Operation Oblong  i2mo,  *i  50 

Desaint,  A.     Three  Hundred  Shades  and  How  to  Mix  Them 8vo,  *io  oo 

Dibdin,  W.  J.     Public  Lighting  by  Gas  and  Electricity 8vo,  *8  oo 

—  Purification  of  Sewage  and  Water 8vo,  6  50 

Dieterich,  K.     Analysis  of  Resins,  Balsams,  and  Gum  Resins 8vo,  *3  oo 

Dinger,  Lieut.  H.  C.     Care  and  Operation  of  Naval  Machinery i2mo,  *2  oo 

Dixon,  D.  B.     Machinist's  and  Steam  Engineer's  Practical  Calculator. 

i6mo,  morocco,  i  25 

Doble,  W.  A.     Power  Plant  Construction  on  the  Pacific  Coast  (In  Press.) 
Dodd,  G.     Dictionary    of    Manufactures,    Mining,    Machinery,    and    the 

Industrial  Arts i2mo,  i  50 

Dorr,  B.  F.     The  Surveyor's  Guide  and  Pocket  Table-book. 

i6mo,  morocco,  2  oo 

Down,  P.  B.     Handy  Copper  Wire  Table , i6mo,  *i  oc 

Draper,  C.  H.     Elementary  Text-book  of  Light,  Heat  and  Sound. .  .  i2mo,  i  oo 

—  Heat  and  the  Principles  of  Thermo-dynamics i2mo,  i  50 

Duckwall,  E.  W.     Canning  and  Preserving  of  Food  Products 8vo,  *5  oo 

Dumesny,  P.,  and  Noyer,  J.     Wood  Products,  Distillates,  and  Extracts. 

8vo,  *4  50 
Duncan,  W.  G.,  and  Penman,  D.     The  Electrical  Equipment  of  Collieries. 

8vo,  *4  oo 


D.  VAN   NOSTRAND   COMPANY'S   SHORT   TITLE    CATALOG        7 

Duthie,  A.  L.     Decorative  Glass  Processes.     (Westminster  Series.)-  .  8vo,  *2  oo 

Dyson,  S.  S.     Practical  Testing  of  Raw  Materials 8vo,  *5  oo 

Eccles,  R.  G.,  and  Duckwall,  E.  W.     Food  Preservatives 8vo,  i  oo 

Paper o  50 

Eddy,  H.  T.     Researches  in  Graphical  Statics 8vo,  i  50 

—  Maximum  Stresses  under  Concentrated  Loads 8vo,  i  50 

Edgcumbe,  K.     Industrial  Electrical  Measuring  Instruments 8vo,  *2  50 

Eissler,  M.     The  Metallurgy  of  Gold 8vc,  7  50 

-  The  Hydrometallurgy  of  Copper 8vo,  *4  50 

-  The  Metallurgy  of  Silver 8vo,  4  oo 

-  The  Metallurgy  of  Argentiferous  Lead 8vo,  5  oo 

—  Cyanide  Process  for  the  Extraction  of  Gold 8vo,  3  oo 

—  A  Handbook  on  Modern  Explosives 8vo,  5  oo 

Ekin,  T.  C.     Water  Pipe  and  Sewage  Discharge  Diagrams folio,  *3  oo 

Eliot,  C.  W.,  and  Storer,  F.  H.     Compendious  Manual  of  Qualitative 

Chemical  Analysis i2mo,  *i  25 

Elliot,  Major  G.  H.     European  Light-house  Systems 8vo,  5  oo 

Ennis,  Wm.  D.     Linseed  Oil  and  Other  Seed  Oils 8vo,  *4  oo 

—  Applied  Thermodynamics 8vo  (In  Press.} 

Erfurt,  J.     Dyeing  of  Paper  Pulp.     Trans,  by  J.  Hubner 8vo,  *7  50 

Erskine-Murray,  J.     A  Handbook  of  Wireless  Telegraphy 8vo,  *3  50 

Evans,  C.  A.     Macadamized  Roads (In  Press.) 

Ewing,  A.  J.     Magnetic  Induction  in  Iron 8vo,  *4  oo 

Fairie,  J.     Notes  on  Lead  Ores i2mo,  *i  oo 

-  Notes  on  Pottery  Clays i2mo,  *i  50 

Fairweather,  W.  C.     Foreign  and  Colonial  Patent  Laws 8vo,  *3  oo 

Fanning,  J.  T.     Hydraulic  and  Water-supply  Engineering 8vo,  *5  oo 

Fauth,  P.     The  Moon  in  Modern   Astronomy.     Trans,  by  J.  McCabe. 

8vo,  *2  oo 

Fay,  I.  W.     The  Coal-tar  Colors 8vo  (In  Press.) 

Fernbach,  R.  L.     Glue  and  Gelatine 8vo,  *3  oo 

Fischer,  E.     The  Preparation  of  Organic  Compounds.     Trans,  by  R.  V. 

Stanford i2mo,  *i  25 

Fish,  J.  C.  L.     Lettering  of  Working  Drawings Oblong  8vo,  i  oo 

Fisher,  H.  K.  C.,  and  Darby,  W.  C.     Submarine  Cable  Testing 8vo,  *3  50 

Fiske,  Lieut.  B.  A.     Electricity  in  Theory  and  Practice 8vo,  2  50 

Fleischmann,  W.    The  Book  of  the  Dairy.  Trans,  by  C.  M.  Aikman.   8vo,  4  oo 
Fleming,    J.    A.     The    Alternate-current    Transformer.     Two   Volumes. 

8vo. 

Vol.    I.     The  Induction  of  Electric  Currents *5  oo 

Vol.  II.     The  Utilization  of  Induced  Currents *S  oo 

Centenary  of  the  Electrical  Current 8vo,  *o  50 

Electric  Lamps  and  Electric  Lighting 8vo,  *3  oo 

Electrical  Laboratory  Notes  and  Forms 4to,  *5  oo 

—  A  Handbook  for  the  Electrical  Laboratory  and  Testing  Room.     Two 

Volumes 8vo,  each,  *5  oo 

Fluery,  H.     The  Calculus  Without  Limits  or  Infinitesimals.     Trans,  by 

C.  O.  Mailloux (In  Prew.) 

Foley,  N.     British  and  American  Customary  and  Metric  Measures .  .  folio,  *3  oo 


8        D.    VAN   NOSTRAND   COMPANY'S   SHORT  TITLE   CATALOG 

Foster,  H.  A.     Electrical  Engineers'  Pocket-book.     (Sixth  Edition.) 

i2mo,  leather,  5  oo 

Foster,  Gen.  J.  G.     Submarine  Blasting  in  Boston  (Mass.)  Harbor..  .  .4to,  3  50 

Fowle,  F.  F.     Overhead  Transmission  Line  Crossings 121110,  *i  50 

-  The  Solution  of  Alternating  Current  ^Problems 8vo  (In  Press.) 

Fox,  W.,  and  Thomas,  C.  W.     Practical  Course  in  Mechanical  Draw- 
ing  i2mo,  i  25 

Francis,  J.  B.     Lowell  Hydraulic  Experiments 4to,  15  oo 

Fuller,  G.  W.     Investigations  into  the  Purification  of  the  Ohio  River. 4to,  *io  oo 

Furnell,  J.     Paints,  Colors,  Oils,  and  Varnishes 8vo,  *i  oo 

Gant,  L.  W.  Elements  of  Electric  Traction 8vo,  *2  50 

Garcke,  E.,  and  Fells,  J.  M.  Factory  Accounts 8vo,  3  oo 

Garforth,  W.  E.  Rules  for  Recovering  Coal  Mines  after  Explosions  and 

Fires i2mo,  leather,  i  50 

Geerligs,  H.  C.  P.  Cane  Sugar  and  Its  Manufacture 8vo,  *s  oo 

Geikie,  J.  Structural  and  Field  Geology 8vo,  *4  oo 

Gerber,  N.  Analysis  of  Milk,  Condensed  Milk,  and  Infants'  Milk-Food.  8vo,  i  25 
Gerhard,  W.  P.  Sanitation,  Watersupply  and  Sewage  Disposal  of  Country 

Houses i2mo,  *2  oo 

Gerhardi,  C.  W.  H.  Electricity  Meters 8vo,  *4  oo 

Geschwind,  L.  Manufacture  of  Alum  and  Sulphates.  Trans,  by  C. 

Salter 8vo,  *5  oo 

Gibbs,  W.  E.  Lighting  by  Acetylene .  i2mo,  *i  50 

-  Physics  of  Solids  and  Fluids.     (Carnegie  Technical  School's  Text- 

books.)   *i  50 

Gibson,  A.  H.     Hydraulics  and  Its  Application 8vo,  *5  oo 

-  Water  Hammer  in  Hydraulic  Pipe  Lines i2mo,  *2  oo 

Gillmore,  Gen.  Q.  A.     Limes,  Hydraulic  Cements  and  Mortars 8vo,  4  oo 

—  Roads,  Streets,  and  Pavements ~~.  .  i2mo,  2  oo 

Golding,  H.  A.     The  Theta-Phi  Diagram i2mo,  *i  25 

Goldschmidt,  R.     Alternating  Current  Commutator  Motor 8vo,  *3  oo 

Goodchild,  W.     Precious  Stones.     (Westminster  Series.) 8vo,  *2  oo 

Goodeve,  T.  M.     Textbook  on  the  Steam-engine i2mo,  2  oo 

Gore,  G.     Electrolytic  Separation  of  Metals 8vo,  *3  50 

Gould,  E.  S.     Arithmetic  of  the  Steam-engine i2mo,  i  oo 

—  Practical  Hydrostatics  and  Hydrostatic  Formulas.     (Science  Series.) 

1 6  no,  o  50 
Grant,  J.     Brewing  and  Distilling.     (Westminster  Series.)  8vo  (In  /V^.s.s.) 

Gray,  J.     Electrical  Influence  Machines ^ i2mo,  2  oo 

Greenwood,  E.     Classified  Guide  to  Technical  and  Commercial  Books.  8vo,  *3  oo 

Gregorius,  R.     Mineral  Waxes.     Trans,  by  C.  Salter i2mo,  *3  oo 

Griffiths,  A.  B.     A  Treatise  on  Manures i2mo,  3  or 

—  Dental  Metallurgy 8vo,  *3  54 

Gross,  E.'    Hops 8vo,  *4  50 

Grossman,  J.     Ammonia  and  Its  Compounds i2ino,  *i  25 

Groth,  L.  A.     Welding  and  Cutting  Metals  by  Gases  or  Electricity.  .  .  .8vo,  *3  oo 

Grover,  F.     Modern  Gas  and  Oil  Engines 8vo,  *2  oo 

Gruner,  A.     Power-loom  Weaving 8vo,  *3  oo 

Guldner,  Hugo.     Internal  Combustion  Engines.     Trans,  by  H.  Diederichs. 

4to,  *io  oo 


D.   VAN   NOSTRAND   COMPANY'S   SHORT  TITLE   CATALOG        9 

Gunther,  C.  0.     Integration i2mo,  *i  25 

Gurden,  R.  L.     Traverse  Tables folio,  half  morocco,  7  50 

Guy,  A.  E.     Experiments  on  the  Flexure  of  Beams 8vo,  *i  25 

Haeder,    H.      Handbook    on    the    Steam-engine.      Trans,  by   H.  H.  P. 

Powles i2mo,  3  oo 

Hainbach,  R.     Pottery  Decoration.     Trans,  by  C.  Slater i2mo,  *3  oo 

Hale,  W.  J.     Calculations  of  General  Chemistry 12010,  *i  oo 

Hall,  C.  H.     Chemistry  of  Paints  and  Paint  Vehicles i2mo,  *2  oo 

Hall,  R.  H.     Governors  and  Governing  Mechanism i2mo,  *2  oo 

Hall,  W.  S.     Elements  of  the  Differential  and  Integral  Calculus 8vo,  *2  25 

—  Descriptive  Geometry 8vo  volume  and  a  4to  atlas,  *3  50 

Haller,  G.  F.,  and  Cunningham,  E.  T.     The  Tesla  Coil (In  Press.) 

Halsey,  F.  A.     Slide  Valve  Gears i2mo,  i  50 

-  The  Use  of  the  Slide  Rule.     (Science  Series.) i6mo,  o  50 

-  Worm  and  Spiral  Gearing.     (Science  Series.) i6mo,  o  50 

Hamilton,  W.  G.     Useful  Information  for  Railway  Men i6mo,  i  oo 

Hammer,  W.  J.     Radium  and  Other  Radio-active  Substances 8vo,  *i  oo 

Hancock,  H.     Textbook  of  Mechanics  and  Hydrostatics 8vo,  i  50 

Hardy,  E.     Elementary  Principles  of  Graphic  Statics i2mo,  *i  50 

Harper,  W.  B.     Utilization  of  Wood  Waste  by  Distillation 4to,  *3  oo 

Harrison,  W.  B.     The  Mechanics'  Tool-book .  *  .  .  i2mo,  i  50 

Hart,  J.  W.     External  Plumbing  Work 8vo,  *3  oo 

—  Hints  to  Plumbers  on  Joint  Wiping „ 8vo,  *3  oo 

—  Principles  of  Hot  Water  Supply 8vo,  *3  oo 

—  Sanitary  Plumbing  and  Drainage 8vo,  *3  oo 

Haskins,  C.  H.     The  Galvanometer  and  Its  Uses i6mo,  i  50 

Hatt,  J.  A.  H.     The  Colorist square  i2mo,  *i  50 

Hausbrand,  E.     Drying  by  Means  of  Air  and  Steam.     Trans,  by  A.  C. 

Wright i2mo,  *2  oo 

—  Evaporating,  Condensing  and  Cooling  Apparatus.     Trans,  by  A.  C. 

Wright 8vo,  *5  oo 

Hausner,  A.     Manufacture  of  Preserved  Foods  and  Sweetmeats.     Trans. 

by  A.  Morris  and  H.  Robson 8vo,  *3  oo 

Hawke,  W.  H.     Premier  Cipher  Telegraphic  Code 4to,  *5  oo 

100,000  Words  Supplement  to  the  Premier  Code 4to,  *5  oo 

Hawkesworth,  J.     Graphical  Handbook  for  Reinforced  Concrete  Design. 

4to,  *2  50 

Hay,  A.     Alternating  Currents 8vo,  *2  50 

—  Principles  of  Alternate-current  Working i2mo,  2  oo 

—  Electrical  Distributing  Networks  and  Distributing  Lines 8vo,  *3  50 

—  Continuous  Current  Engineering 8vo,  *2  50 

Heap,  Major  D.  P.     Electrical  Appliances 8vo,  2  oo 

Heaviside,  0.     Electromagnetic  Theory.     Two  Volumes 8vo,  each,  *5  oo 

Heck,  R.  C.  H.     Steam-Engine  and  Other  Steam  Motors.     Two  Volumes. 

Vol.    I.     Thermodynamics  and  the  Mechanics 8vo,  *3  50 

Vol.  II.     Form,  Construction,  and  Working 8vo,  *5  oo 

Abridged  edition  of  above  volumes  (Elementary) 8vo  (In  Preparation.) 

Notes  on  Elementary  Kinematics 8vo^  boards,  *i  oo 

-  Graphics  of  Machine  Forces 8vo,  boards,'  *i  oo 

Hedges,  K.     Modern  Lightning  Conductors 8vo,  3  oo 


10     D.  VAN  NOSTRAND  COMPANY'S  SHORT  TITLE  CATALOG 

Heermann,  P.     Dyers'  Materials.     Trans,  by  A.  C.  Wright i2mo,  *2  50 

Hellot,  Macquer  and  D'Apligny.     Art  of  Dyeing  Wool,  Silk  and  Cotton. 

8vo,  *2  oo 

Henrici,  0.     Skeleton  Structures 8vo,  i  50 

Hermann,  F.     Painting  on  Glass  and  Porcelain 8vo,  *3  50 

Herrmann,  G.     The  Graphical  Statics  of  Mechanism.     Trans,  by  A.  P. 

Smith i2mo,  2  oo 

Herzfeld,  J.     Testing  of  Yarns  and  Textile  Fabrics 8vo,  *3  50 

Hildebrandt,  A.     Airships,  Past  and  Present 8vo,  *3  50 

Hill,  J.  W.     The  Purification  of  Public  Water  Supplies.      New  Edition.     (In  Press.) 

—  Interpretation  of  Water  Analysis (In  Press.) 

Hiroi,  I.     Statically-Indeterminate  Stresses i2mo,  *2  oo 

Hirshfeld,  C.  F.     Engineering  Thermodynamics.     (Science  Series.).  i6mo,  050 

Hobart,  H.  M.     Heavy  Electrical  Engineering 8vo,  *4  50 

—  Electricity 8vo,  *2  oo 

—  Electric  Trains (In  Pre.v.s.) 

Hobbs,  W.  R.  P.     The  Arithmetic  of  Electrical  Measurements i2mo,  o  50 

Hoff,  J.  N.     Paint  and  Varnish  Facts  and  Formulas i2ino,  *3  oo 

Hoff,  Com.  W.  B.     The  Avoidance  of  Collisions  at  Sea.  .  .  i6mo,  morocco,  o  75 

Hole,  W.     The  Distribution  of  Gas 8vo,  *y  50 

Holley,  A.  L.     Railway  Practice folio,  12  oo 

Holmes,  A.  B.     The  Electric  Light  Popularly  Explained  ....  i2mo,  paper,  o  50 

Hopkins,  N.  M.     Experimental  Electrochemistry 8vo,  *3  oo 

—  Model  Engines  and  Small  Boats „ i2mo,  i  25 

Horner,  J.     Engineers'  Turning 8vo,  *3  50 

—  Metal  Turning i2mo,  i  50 

-  Toothed  Gearing i2mo,  2  25 

Houghton,  C.  E.     The  Elements  of  Mechanics  of  Materials i2mo,  *2  oo 

Houllevique,  L.     The  Evolution  of  the  Sciences 8vo,  *2  oo 

Howe,  G.     Mathematics  for  the  Practical  Man 12010  (In  Press.} 

Howorth,  J.     Repairing  and  Riveting  Glass,  China  and  Earthenware. 

8vo,  paper,  *o  50 

Hubbard,  E.     The  Utilization  of  Wood- waste 8vo,  *2  50 

Humber,  W.     Calculation  of  Strains  in  Girders i2mo,  2  50 

Humphreys,  A.  C.     The  Business  Features  of  Engineering  Practice.  .  8vo,  *i  25 

Hurst,  G.  H.     Handbook  of  the  Theory  of  Color 8vo,  *2  50 

—  Dictionary  of  Chemicals  and  Raw  Products 8vo,  *3  oo 

—  Lubricating  Oils,  Fats  and  Greases 8vo,  *3  oo 

—  Soaps 8vo,  *s  oo 

-  Textile  Soaps  and  Oils 8vo,  *2  50 

HutchinsonR.  W.,  Jr.     Long  Distance  Electric  Power  Transmission .  1 2mo,  *3  oo 
Hutchinson,  R.  W.,  Jr.,  and  Ihlseng,  M.  C.     Electricity  in  Mining.  .  i2mo, 

(In  7Vr.s-.s-) 
Hutchinson,  W.  B.     Patents  and  How  to  Make  Money  Out  of  Them. 

i2mo,  i  25 

Hutton,  W.  S.     Steam-boiler  Construction 8vo,  6  oo 

—  Practical  Engineer's  Handbook 8vo,  7  oo 

—  The  Works'  Manager's  Handbook 8vo,  6  oo 

Ingle,  H.     Manual  of  Agricultural  Chemistry 8vo,  *3  oo 

Innes,  C,  H.     Problems  in  Machine  Design i2mo,  *2  oo 


D.   VAN   NOSTRAND   COMPANY'S   SHORT  TITLE  CATALOG      11 

Innes,  C.  H.     Air  Compressors  and  Blowing  Engines 12010,  *2  oo 

—  Centrifugal  Pumps i2mo,  *2  oo 

—  The  Fan i2mo,  *2  oo 

Isherwood,  B.  F.     Engineering  Precedents  for  Steam  Machinery 8vo,  2  50 

\ 

Jamieson,  A.     Text  Book  on  Steam  and  Steam  Engines 8vo,  3  oo 

Elementary  Manual  on  Steam  and  the  Steam  Engine i2mo,  i  50 

Jannettaz,  E.     Guide  to  the  Determination  of  Rocks.     Trans,  by  G.  W. 

Plympton i2mo,  i  50 

Jehl,  F.     Manufacture  of  Carbons 8vo,  *4  oo 

Jennings,  A.  S.     Commercial  Paints  and  Painting.     (Westminster  Series.) 

8vo  {In  Press.) 

Jennison,  F.  H.     The  Manufacture  of  Lake  Pigments 8vo,  *3  oo 

Jepson,  G.     Cams  and  the  Principles  of  their  Construction 8vo,  *i  50 

-   Mechanical  Drawing 8vo  (In  Preparation.) 

Jockin,  W.     Arithmetic  of  the  Gold  and  Silversmith i2mo,  *i  oo 

Johnson,  G.  L.     Photographic  Optics  and  Color  Photography. 8vo,  *3  oo 

Johnson,  W.  H.     The  Cultivation  and  Preparation  of  Para  Rubber. .  .8vo,  *3  oo 

Johnson,  W.  McA.     The  Metallurgy  of  Nickel (In  Preparation.) 

Johnston,  J.  F.  W.,  and  Cameron,  C.     Elements  of  Agricultural  Chemistry 

and  Geology i2mo,  2  60 

Joly,  J.     Raidoactivity  and  Geology i2mo,  *3  oo 

Jones,  H.  C.     Electrical  Nature  of  Matter  and  Radioactivity i2mo,  2  oo 

Jones,  M.  W.     Testing  Raw  Materials  Used  in  Paint i2mo,  *2  oo 

Jones,  L.,  and  Scard,  F.  I.     Manufacture  of  Cane  Sugar 8vo,  *5  oo 

Joynson,  F.  H.     Designing  and  Construction  of  Machine  Gearing. .  .  .8vo,  2  oo 

Juptner,  H.  F.  V.     Siderology:  The  Science  of  Iron 8vo,  *5  o. 

Kansas  City  Bridge \  .  .  .  .  4to,  6  oo 

Kapp,  G.     Electric  Transmission  of  Energy I2mo,  3  50 

—  Dynamos,  Motors,  Alternators  and  Rotary  Converters.     Trans,  by 

H.  H.  Simmons 8vo,  4  oo 

Keim,  A.  W.     Prevention  of  Dampness  in  Buildings 8vo,  *2  oo 

Keller,  S.  S.     Mathematics  for  Engineering  Students.     i2mo,  half  leather. 

Algebra  and  Trigonometry,  with  a  Chapter  on  Vectors *i  75 

Special  Algebra  Edition *i  oo 

Plane  and  Solid  Geometry *i  25 

Analytical  Geometry  and  Calculus. *2  oo 

Kelsey,  W.  R.     Continuous-current  Dynamos  and  Motors 8vo,  *2  50 

Kemble,  W.  T.,  and  Underbill,  C.  R.     The  Periodic  Law  and  the  Hydrogen 

Spectrum .  .8vo,  paper,  *o  50 

Kemp,  J.  F.     Handbook  of  Rocks 8vo,  *i  50 

Kendall,  E.     Twelve  Figure  Cipher  Code 4to,  *is  oo 

Kennedy,  R.     Modern  Engines  and  Power  Generators.     Six  Volumes.    4to,  1500 

Single  Volumes each,  3  oo 

—  Electrical  Installations.     Five  Volumes 4to,  15  oo 

Single  Volumes each,  3  50 

—  Flying  Machines;  Practice  and  Design i2mo,  *2  oo 

Kennelly,  A.  E.     Electro-dynamic  Machinery 8vo,  i  50 

Kershaw,  J.  B.  C.     Fuel,  Water  and  Gas  Analysis 8vo,  *2  50 

—  Electrometallurgy.     (Westminster  Series.) 8vo,  *2  oo 


12      D.   VAN   NOSTRAND  COMPANY'S   SHORT  TITLE   CATALOG 
Kershaw,  J.  B.  C.     The  Electric  Furnace  in  Iron  and  Steel  Production. 

I2H1O,  *I    50 

Kingdon,  J.  A.     Applied  Magnetism 8vo,  *3  oo 

Kinzbrunner,  C.     Alternate  Current  Windings 8vo,  *i  50 

—  Continuous  Current  Armatures 8vo,  *i  50 

—  Testing  of  Alternating  Current  Machines 8vo,  *2  oo 

Kirkaldy,  W.  G.     David  Kirkaldy's  System  of  Mechanical  Testing.  . .  .4to,  10  oo 

Kirkbride,  J.     Engraving  for  Illustration 8vo,  *i  50 

Kirkwood,  J.  P.     Filtration  of  River  Waters .4to,  7  50 

Klein,  J.  F.     Design  of  a  High-speed  Steam-engine 8vo,  *5  oo 

Kleinhans,  F.  B.     Boiler  Construction 8vo,  3  oo 

Knight,  Lieut.-Com.  A.  M.     Modern  Seamanship 8vo,  *6  oo 

Half  morocco *7  50 

Knox,  W.  F.     Logarithm  Tables (In  Preparation.) 

Knott,  C.  G.,  and  Mackay,  J.  S.     Practical  Mathematics 8vo,  2  oo 

Koester,  F.     Steam-Electric  Power  Plants 4to,  *5  oo 

—  Hydroelectric  Developments  and  Engineering 4to,  *s  oo 

Koller,  T.     The  Utilization  of  Waste  Products 8vo,  *3  50 

—  Cosmetics : 8vo,  *2  50 

Krauch,  C.     Testing  of  Chemical  Reagents.     Trans,  by  J.  A.  Williamson 

and  L.  W.  Dupre 8vo,  *3  oo 

Lambert,  T.     Lead  and  its  Compounds 8vo,  *3  50 

—  Bone  Products  and  Manures 8vo,  *3  oo 

Lamborn,  L.  L.     Cottonseed  Products 8vo,  *3  oo 

—  Modern  Soaps,  Candles,  and  Glycerin 8vo,  *7  50 

Lamprecht,  R.     Recovery  Work  After  Pit  Fires.     Trans,  by  C.  Salter .  .  8vo,  *4  oo 
Lanchester,  F.  W.     Aerial  Flight.     Two  Volumes.     8vo. 

Vol.    I.     Aerodynamics *6  oo 

Vol.  II.     Aerodonetics *6  oo 

Larner,  E.  T.     Principles  of  Alternating  Currents i2mo,  *i  25 

Larrabee,  C.  S.     Cipher  and  Secret  Letter  and  Telegraphic  Code.  . .  .  i6mo,  o  60 
Lassar-Cohn,  Dr.     Modern  Scientific  Chemistry.     Trans,  by  M.  M.  Patti- 

son  Muir i2mo,  *2  oo 

Latta,  M.  N.     Handbook  of  American  Gas-Engineering  Practice 8vo,  *4  50 

—  American  Producer  Gas  Practice 4to,  *6  oo 

Leask,  A.  R.     Breakdowns  at  Sea i2mo,  2  oo 

-  Triple  and  Quadruple  Expansion  Engines i2mo,  2  oo 

—  Refrigerating  Machinery i2mo,  2  oo 

Lecky,  S.  T.  S.     "  Wrinkles  "  in  Practical  Navigation 8vo,  *8  oo 

Leeds,  C.  C.     Mechanical  Drawing  for  Trade  Schools oblong  4to, 

High  School  Edition *i   25 

Machinery  Trades  Edition *2  oo 

LefeVre,  L.     Architectural  Pottery.      Trans,  by  H.  K.  Bird  and  W.  M. 

Binns •. 4to,  *7  50 

Lehner,  S.     Ink  Manufacture.     Trans,  by  A.  Morris  and  H.  Robson  .  .8vo,  *2  50 

Lemstrom,  S.     Electricity  in  Agriculture  and  Horticulture Svo,  *i  50 

Lewes,  V.  B.     Liquid  and  Gaseous  Fuels.     (Westminster  Series.).  .  .  .8vo,  *2  oo 

Lieber,  B.  F.     Lieber's  Standard  Telegraphic  Code Svo,  *io  oo 

—  Code.     German  Edition Svo,  *io  oo 

—  Spanish  Edition Svo,  *io  oo 


D.   VAN   NOSTRAND   COMPANY'S  SHORT  TITLE  CATALOG      13 

Lieber,  B.  F.     Code.     French  Edition .  .8vo,  *io  oo 

-  Terminal  Index 8vo,     *2  50 

—  Lieber's  Appendix folio,  *i$  oo 

—  Handy  Tables 4to,     *2  50 

—  Bankers  and  Stockbrokers'  Code  and  Merchants  and  Shippers'  Blank 

Tables 8vo,  *is  oo 

—  100,000,000  Combination  Code 8vo,  *i$  oo 

—  Engineering  Code 8vo,  *io  oo 

Livermore,  V.  P.,  and  Williams,  J.     How  to  Become  a  Competent  Motor- 
man i2mo,  *i  oo 

Livingstone,  R.     Design  and  Construction  of  Commutators 8vo,  *2  25 

Lobben,  P.     Machinists'  and  Draftsmen's  Handbook 8vo,  2  50 

Locke,  A.  G.  and  C.  G.     Manufacture  of  Sulphuric  Acid 8vo,  10  oo 

Lockwood,  T.  D.     Electricity,  Magnetism,  and  Electro-telegraphy. .  .  .  8vo,  2  50 

—  Electrical  Measurement  and  the  Galvanometer I2mo,  i  50 

Lodge,  0.  J.     Elementary  Mechanics ' i2mo,  i  50 

—  Signalling  Across  Space  without  Wires 8vo,  *2  oo 

Lord,  R.  T.     Decorative  and  Fancy  Fabrics 8vo,  *3  50 

Loring,  A.  E.     A  Handbook  of  the  Electromagnetic  Telegraph i6mo,  o  50 

Lowenstein,  L.  C.,  and  Crissey,  C.  P.     Centrifugal  Pumps.  .  .  .  (In  Press.) 

Lucke,  C.  E.     Gas  Engine  Design 8vo,  *3  oo 

—  Power  Plants:  their  Design,  Efficiency,  and  Power  Costs.     2  vols. 

(In  Preparation.) 

—  Power  Plant  Papers.     Form  I.  The  Steam  Power  Plant paper,     *i  50 

Lunge,  G.     Coal-tar  and  Ammonia.     Two  Volumes 8vo,  *i$  oo 

—  Manufacture  of  Sulphuric  Acid  and  Alkali.     Three  Volumes.  .  .  .8vo, 

Vol.     I.     Sulphuric  Acid.     In  two  parts *i5  oo 

Vol.    II.     Salt  Cake,  Hydrochloric  Acid  and  Leblanc  Soda.      In  two 

parts *i5  oo 

Vol.  III.     Ammonia  Soda *i5  oo 

-  Technical  Chemists'  Handbook i2mo,  leather,     *3  50 

—  Technical  Methods  of  Chemical  Analysis.     Trans,  by  C.  A.  Keane. 

in  collaboration  with  the  corps  of  specialists. 

Vol.    I.     In  two  parts. . .' 8vo,  *i$  oo 

Vols.  II  and  III (In  Preparation.) 

Lupton,  A.,  Parr,  G.  D.  A.,  and  Perkin,  H.     Electricity  as  Applied  to 

Mining 8vo,     *4  50 

Luquer,  L.  M.     Minerals  in  Rock  Sections 8vo,     *i  50 

Macewen,  H.  A.     Food  Inspection 8vo,  *2  50 

Mackie,  J.     How  to  Make  a  Woolen  Mill  Pay 8vo,  *2  oo 

Mackrow,  C.     Naval  Architect's  and  Shipbuilder's  Pocket-book. 

i6mo,  leather,  5  oo 

Maguire,  Capt.  E.     The  Attack  and  Defense  of  Coast  Fortifications.  .  .8vo,  2  50 

Maguire,  Wm.  R.     Domestic  Sanitary  Drainage  and  Plumbing 8vo,  4  oo 

Marks,  E.  C.  R.     Construction  of  Cranes  and  Lifting  Machinery.  .  .  .  i2mo,  *i  50 

—  Construction  and  Working  of  Pumps i2mo,  *i  50 

—  Manufacture  of  Iron  and  Steel  Tubes I2mo,  *2  oo 

-  Mechanical  Engineering  Materials i2mo,  *i  oo 

Marks,  G.  C.     Hydraulic  Power  Engineering 8vo,  3  50 

—  Inventions,  Patents  and  Designs I2mo,     *i  oo 


14     D.  VAN   NOSTRAND   COMPANY'S   SHORT  TITLE   CATALOG 

Markham,  E.  R.     The  American  Steel  Worker i2mo,  2  50 

Marlow,  T.  G.     Drying  Machinery  and  Practice 8vo, 

Marsh,  C.  F.     Concise  Treatise  on  Reinforced  Concrete 8vo,  *2  50 

Marsh,  C.  F.,  and  Dunn,  W,     Reinforced  Concrete 4to,  *5  oo 

—  Manual  of  Reinforced  Concrete  and  Concrete  Block  Construction. 

i6mo,  morocco,  *2  50 
Massie,  W.  W.,  and  Underbill,  C.  R.     Wireless  Telegraphy  and  Telephony. 

i2mo,  *i  oo 
Matheson,  D.     Australian  Saw-Miller's  Log  and  Timber  Ready  Reckoner. 

i2mo,  leather,  i  50 

Mathot,  R.  E.     Internal  Combustion  Engines 8vo, 

Maurice,  W.     Electric  Blasting  Apparatus  and  Explosives 8vo,  *3  50 

-  Shot  Firer's  Guide 8vo,  *i  50 

Maxwell,  W.  H.,  and  Brown,  J.  T.     Encyclopedia  of  Municipal  and  Sani- 
tary Engineering 4to,  *io  oo 

Mayer,  A.  M.     Lecture  Notes  on  Physics 8vo,  2  oo 

McCullough,  R.  S.     Mechanical  Theory  of  Heat 8vo,  3  50 

Mclntosh,  J.  G.     Technology  of  Sugar 8vo,  *4  50 

—  Industrial  Alcohol., ... 8vo,  *3  oo 

—  Manufacture  of  Varnishes  and  Kindred  Industries.     Three  Volumes. 

8vo. 

Vol.     I.     Oil  Crushing,  Refining  and  Boiling *3  50 

Vol.    II.     Varnish  Materials  and  Oil  Varnish  Making *4  oo 

VoL'III : -..(In  Preparation.') 

McMechen,  F.  L.     Tests  for  Ores,  Minerals  and  Metals i2mo,  *i  oo 

McNeill,  B.     McNeill's  Code 8vo,  *6  oo 

McPherson,  J.  A.     Water- works  Distribution 8vo,  2  50 

Melick,  C.  W.     Dairy  Laboratory  Guide i2mo,  *i  25 

Merck,  E.     Chemical  Reagents;  Their  Purity  and  Tests 8vo,  *i  50 

Merritt,  Wm.  H.     Field  Testing  for  Gold  and  Silver i6mo,  leather,  i  50 

Meyer,  J.  G.  A.,  and  Pecker,  C.  G.     Mechanical  Drawing  and  Machine 

Design 4to,  5  oo 

Michell,  S.     Mine  Drainage 8vo,  10  oo 

Mierzinski,  S.     Waterproofing  of  Fabrics.     Trans,  by  A.  Morris  and  H. 

Robson 8vo,  *2  50 

Miller,  E.  H.     Quantitative  Analysis  for  Mining  Engineers 8vo,  *i  50 

Milroy,  M.  E.  W.     Home  Lace-making i2mo,  *i  oo 

Minifie,  W.     Mechanical  Drawing 8vo,  *4  oo 

Modern  Meteorology I2mo,  i  50 

Monckton,  C.  C.  F.     Radiotelegraphy.     (Westminster  Series.) 8vo,  *2  oo 

Monteverde,  R.  D.     Vest  Pocket  Glossary  of  English-Spanish,  Spanish- 
English  Technical  Terms 64mo,  leather,  *i  oo 

L'oore,  E.  C.  S.     New  Tables  for  the  Complete  Solution  of  Ganguillet  and 

Kutter's  Formula 8vo,  *5  oo 

Moreing,  C.  A.,  and  Neal,  T.    New  General  and  Mining  Telegraph  Code,  8vo,  *5  oo 

Morgan,  A.  P.     Wireless  Telegraph  Apparatus  for  Amateurs i2mo,  *i  50 

Moses,  A.  J.     The  Characters  of  Crystals 8vo,  *2  oo 

Moses,  A.  J.,  and  Parsons,  C.  L.     Elements  of  Mineralogy 8vo,  *2  50 

Moss,  S.  A.     Elements  of  Gas  Engine  Design.     (Science  Series.) ....  i6mo,  o  50 

—  The  Lay-out  of  Corliss  Valve  Gears.     (Science  Series.) i6mo,  o  50 

Mullin,  J.  P.     Modern  Moulding  and  Pattern- making i2mo,  2  50 


D.  VAN  NOSTRAND   COMPANY'S   SHORT  TITLE  CATALOG      15 

Munby,  A.  E.     Chemistry  and  Physics  of  Building  Materials.     (Westmin- 
ster Series.) 8vo,  *2  oo 

Murphy,  J.  G.     Practical  Mining : i6mo,  i  oo 

Murray,  J.  A.     Soils  and  Manures.     (Westminster  Series.) 8vo,  *2  oo 

Naquet,  A.     Legal  Chemistry I2mo,  2  oo 

Nasmith,  J.     The  Student's  Cotton  Spinning 8vo,  3  oo 

Nerz,  F.     Searchlights.     Trans,  by  C.  Rodgers 8vo,  *3  oo 

Neuberger,  H.,  and  Noalhat,  H.     Technology  of  Petroleum.     Trans,  by  J. 

G.  Mclntosh 8vo,  *io  oo 

Newall,  J.  W.     Drawing,  Sizing  and  Cutting  Bevel-gears 8vo,  i  50 

Newlands,  J.     Carpenters  and  Joiners'  Assistant folio,  half  morocco,  15  oo 

Nicol,  G.     Ship  Construction  and  Calculations 8vo,  *4  50 

Nipher,  F.  E.     Theory  of  Magnetic  Measurements i2mo,  i  oo 

Nisbet,  H.     Grammar  of  Textile  Design 8vo,  *3  oo 

Noll,  A.     How  to  Wire  Buildings i2mo,  i  50 

Nugent,  E.     Treatise  on  Optics i2mo,  i  50 

O'Connor,  H.     The  Gas  Engineer's  Pocketbook i2mo,  leather,  3  50 

Petrol  Air  Gas i2mo,  *o  75 

Olsson,  A.     Motor  Control,  in  Turret  Turning  and  Gun  Elevating.     (U.  S. 

Navy  Electrical  Series,  No.  i.) i2mo,  paper,  *o  50 

Olsen,  J.  C.     Text-book  of  Quantitative  Chemical  Analysis 8vo,  *4  oo 

Oudin,  M.  A.     Standard  Polyphase  Apparatus  and  Systems 8vo,  *3  oo 

Palaz,  A.     Industrial  Photometry.     Trans,  by  G.  W.  Patterson,  Jr. .  .  8vo,  *4  bo 

Pamely,  C.     Colliery  Manager's  Handbook 8vo,  *io  oo 

Parr,  G.  D.  A.     Electrical  Engineering  Measuring  Instruments 8vo,  *3  50 

Parry,  E.  J.     Chemistry  of  Essential  Oils  and  Artificial  Perfumes ....  8vo,  *5  oo 

Parry,  E.  J.,  and  Coste,  J.  H.     Chemistry  of  Pigments 8vo,  *4  50 

Parry,  L.  A.     Risk  and  Dangers  of  Various  Occupations 8vo,  *3  oo 

Parshall,  H.  F.,  and  Hobart,  H.  M.     Armature  Windings 4to,  *7  50 

—  Electric  Railway  Engineering 4to,  *io  oo 

Parshall,  H.  F.,  and  Parry,  E.     Electrical  Equipment  of  Tramways. .  .  .  (In  Press.) 

Parsons,  S.  J.     Malleable  Cast  Iron • 8vo,  *2  50 

Passmore,  A.  C.     Technical  Terms  Used  in  Architecture 8vo,  *3  50 

Patterson,  D.     The  Color  Printing  of  Carpet  Yarns 8vo,  *3  50 

—  Color  Matching  on  Textiles 8vo,  *3  oo 

—  The  Science  of  Color  Mixing 8vo,  *3  oo 

Patton,  H.  B.     Lecture  Notes  on  Crystallography 8vo,  *i  25 

Paulding,  C.  P.     Condensation  of  Steam  in  Covered  and  Bare  Pipes.  .8vo,  *2  oo 

—  Transmission  of  Heat  through  Cold-storage  Insulation i2mo,  *i  oo 

Peirce,  B.     System  of  Analytic  Mechanics 4to,  10  oo 

Pendred,  V.     The  Railway  Locomotive.     (Westminster  Series.) 8vo,  *2  oo 

Perkin,  F.  M.     Practical  Methods  of  Inorganic  Chemistry i2mo,  *i  oo 

Perrigo,  O.  E.     Change  Gear  Devices 8vo,  i  oo 

Perrine,  F.  A.  C.     Conductors  for  Electrical  Distribution 8vo,  *3  50 

Petit,  G.     White  Lead  and  Zinc  White  Paints 8vo,  *i   50 

Petit,  R.     How  to  Build  an  Aeroplane.     Trans,  by  T.  O'B.  Hubbard,  and 

J.  H.  Ledeb-oer 8vo,  *i  50 

Perry,  J.     Applied  Mechanics 8vo>  *2  50 


16      D.   VAN   NOSTRAND   COMPANY'S  SHORT  TITLE   CATALOG 

Phillips,  J.     Engineering  Chemistry 8vo,  *4  50 

—  Gold  Assaying 8vo,  *2  50 

Phin,  J.     Seven  Follies  of  Science ' i2mo,  *i   25 

—  Household  Pests,  and  How  to  Get  Rid  of  Them 8vo  (In  Preparation.) 

Pickworth,  C.  N.     The  Indicator  Handbook.     Two  Volumes.  .  i2mo,  each,  i   50 

—  Logarithms  for  Beginners i2mo,  boards,  o  50 

—  The  Slide  Rule I2mo,  i  oo 

Plane  Table,  The 8vo,  2  oo 

Plattner's  Manual  of  Blow-pipe  Analysis.    Eighth  Edition,  revised.    Trans. 

by  H.  B.  Cornwall .8vo,  *4  oo 

Plympton,  G.  W.     The  Aneroid  Barometer.     (Science  Series.) i6mo,  050 

Pocket  Logarithms  to  Four  Places.     (Science  Series.) i6mo,  o  50 

Pope,  F.  L.     Modern  Practice  of  the  Electric  Telegraph 8vo,  i  50 

Popple  well,  W.  C.  Elementary  Treatise  on  Heat  and  Heat  Engines.  .  i2mo,  *3  oo 

-  Prevention  of  Smoke 8vo,  *3  50 

-  Strength  of  Materials , 8vo,  *i  75 

Potter,  T.     Concrete 8vo,  *3  oo 

Practical  Compounding  of  Oils,  Tallow  and  Grease 8vo,  *3  50 

Practical  Iron  Founding '. i2mo,  i  50 

Pray,  T.,  Jr.     Twenty  Years  with  the  Indicator 8vo,  2  50 

—  Steam  Tables  and  Engine  Constant 8vo,  2  oo 

—  Calorimeter  Tables 8vo,  i  oo 

Preece,  W.  H.     Electric  Lamps (In  Press.} 

Prelini,  C.     Earth  and  Rock  Excavation 8vo,  *3  oo 

—  Graphical  Determination  of  Earth  Slopes 8vo,  *2  oo 

-  Tunneling 8vo,  3  oo 

—  Dredging.     A  Practical  Treatise (In  Press.} 

Prescott,  A.  B.     Organic  Analysis 8vo,  5  oo 

Prescott,  A.  B.,  and  Johnson,  0.  C.     Qualitative  Chemical  Analysis.  .  .8vo,  *3  50 
Prescott,  A.  B.,  and  Sullivan,  E.  C.     First  Book  in  Qualitative  Chemistry. 

i2mo,  *i  50 

Pritchard,  0.  G.  The  Manufacture  of  Electric-light  Carbons .  .  8vo,  paper,  *o  60 
Prost,  E.  Chemical  Analysis  of  Fuels,  Ores,  Metals.  Trans,  by  J.  C. 

Smith 8vo,  *4  50 

Pullen,  W.  W.  F.  Application  of  Graphic  Methods  to  the  Design  of 

Structures i2mo,  *2  50 

—  Injectors:  Theory,  Construction  and  Working • i2mo,  *i  50 

Pulsifer,  W.  H.     Notes  for  a  History  of  Lead 8vo,  4  oo 

Putsch,  A.     Gas  and  Coal-dust  Firing 8vo,  *3  oo 

Pynchon,  T.  R.     Introduction  to  Chemical  Physics 8vo,  3  oo 

Rafter,  G.  W.     Treatment  of  Septic  Sewage.     (Science  Series.) i6mo,  o  50 

Rafter,  G.  W.,  and  Baker,  M.  N.     Sewage  Disposal  in  the  United  States .  4to,  *6  oo 

Raikes,  H.  P.     Sewage  Disposal  Works 8vo,  *4  oo 

Railway  Shop  Up-to-Date 4to,  2  oo 

Ramp,  H.  M.     Foundry  Practice (In  Prr.s.s. ) 

Randall,  P.  M.     Quartz  Operator's  Handbook 12 mo,  2  oo 

Randau,  P.     Enamels  and  Enamelling 8vo,  *4  oo 

Rankine,  W.  J.  M.     Applied  Mechanics 8vo,  5  oo 

—  Civil  Engineering 8vo,  6  50 

-  Machinery  and  Millwork 8vo,  5  oo 


D.   VAN   NOSTRAND   COMPANY'S   SHORT  TITLE  CATALOG      17 

Rankine,  W.  J.  M.     The  Steam-engine  and  Other  Prime  Movers. ....  8vo,  5  oo 

-  Useful  Rules  and  Tables „ 8vo,  4  oo 

Rankine,  W.  J.  M.,  and  Bamber,  E.  F.     A  Mechanical  Text-book..  .  .8vo,  3  50 
Raphael,  F.  C.     Localization  of  Faults  in  Electric  Light  and  Power  Mains. 

8vo,  *3  oo 

Rathbone,  R.  L.  B.     Simple  Jewellery 8vo,  *2  oo 

Rateau,  A.     Flow  of  Steam  through  Nozzles  and  Orifices.      Trans,  by  H. 

B.  Brydon 8vo,  *i  50 

Rausenberger,  F.     The  Theory  of  the  Recoil  of  Guns 8vo,  *4  50 

Rautenstrauch,  W.     Notes  on  the  Elements  of  Machine  Design,  8 vo,  boards,  *i  50 
Rautenstrauch,  W.,  and  Williams,  J.  T.     Machine  Drafting  and  Empirical 
Design. 

Part    I.  Machine  Drafting 8vo,  *i  25 

Part  II.  Empirical  Design (In  Preparation.) 

Raymond,  E.  B.     Alternating  Current  Engineering i2mo,  *2  50 

Rayner,  H.     Silk  Throwing  and  Waste  Silk  Spinning 8vo,  *2  50 

Recipes  for  the  Color,  Paint,  Varnish,  Oil,  Soap  and  Drysaltery  Trades .  8vo,  *3  50 

Recipes  for  Flint  Glass  Making i2mo,  *4  50 

Reed's  Engineers'  Handbook 8vo,  *5  oo 

—  Key  to  the  Nineteenth  Edition  of  Reed's  Engineers'  Handbook .  .  8vo,  *3  oo 

—  Useful  Hints  to  Sea-going  Engineers i2mo,  i  50 

—  Marine  Boilers I2mo,  2  oo 

Reinhardt,  C.  W.     Lettering  for  Draftsmen,  Engineers,  and  Students. 

oblong  4to,  boards,  i  oo 

-  The  Technic  of  Mechanical  Drafting oblong  4to,  boards,  *i  oo 

Reiser,  F.     Hardening  and  Tempering  of  Steel.     Trans,  by  A.  Morris  and 

H.  Robson i2mo,  *2  50 

Reiser,  N.     Faults  in  the  Manufacture  of  Woolen  Goods.     Trans,  by  A. 

Morris  and  H.  Robson 8vo,  *2  50 

—  Spinning  and  Weaving  Calculations 8vo,  *5  oo 

Renwick,  W.  G.     Marble  and  Marble  Working 8vo,  5  oo 

Rhead,  G.  F.     Simple  Structural  Woodwork i2mo,  *i  oo 

Rice,  J.  M.,  and  Johnson,  W.  W.     A  New  Method  of  Obtaining  the  Differ- 
ential of  Functions i2mo,  o  50 

Richardson,  J.     The  Modern  Steam  Engine 8vo,  *3  50 

Richardson,  S.  S.     Magnetism  and  Electricity i2mo,  *2  oo 

Rideal,  S.     Glue  and  Glue  Testing 8vo,  *4  oo 

Rings,  F.     Concrete  in  Theory  and  Practice i2mo,  *2  50 

Ripper,  W.     Course  of  Instruction  in  Machine  Drawing folio,  *6  oo 

Roberts,  J.,  Jr.     Laboratory  Work  in  Electrical  Engineering 8vo,  *2  oo 

Robertson,  L.  S.     Water-tube  Boilers 8vo,  3  oo 

Robinson,  J.  B.     Architectural  Composition 8vo,  *2  50 

Robinson,  S.  W.     Practical  Treatise  on  the  Teeth  of  Wheels.     (Science 

Series.) i6mo,  o  50 

Roebling,  J.  A.     Long  and  Short  Span  Railway  Bridges folio,  25  oo 

Rogers,  A.     A  Laboratory  Guide  of  Industrial  Chemistry i2mo,  *i  50 

Rogers,  A.,  and  Aubert,  A.  B.     Industrial  Chemistry (In  Press.) 

Rollins,  W.     Notes  on  X-Light 8vo,  *7  50 

Rose,  J.     The  Pattern-makers'  Assistant 8vo,  2  50 

—  Key  to  Engines  and  Engine-running i2mo,  2  50 

Rose,  T.  K.     The  Precious  Metals.     (Westminster  Series.) 8vo,  *2  oo 


18     D.  VAN   NOSTRAND   COMPANY'S   SHORT  TITLE   CATALOG 

Rosenhain,  W.     Glass  Manufacture.     (Westminster  Series.) 8vo,  *2  oo 

Rossiter,  J.  T.     Steam  Engines.     (Westminster  Series.). .  .  .8vo  (In  Pms.s.) 
—  Pumps  and  Pumping  Machinery.     (Westminster  Series.).. 8 vo  (In  Prcus.} 

Roth.     Physical  Chemistry 8vo,  *2  oo 

Rouillion,  L.     The  Economics  of  Manual  Training 8vo,  2  oo 

Rowan,  F.  J.     Practical  Physics  of  the  Modern  Steam-boiler 8vo,  7  50 

Roxburgh,  W.     General  Foundry  Practice 8vo,  *3  50 

Ruhmer,  E.     Wireless  Telephony.     Trans,  by  J.  Erskine-Murray ....  8vo,  *3  50 

Russell,  A.     Theory  of  Electric  Cables  and  Networks 8vo,  *3  oo 

Sabine,  R.     History  and  Progress  of  the  Electric  Telegraph i2mo,  i  25 

Saeltzer,  A.     Treatise  on  Acoustics i2mo,  i  oo 

Salomons,  D.     Electric  Light  Installations.     i2mo. 

Vol.    I.     The  Management  of  Accumulators 2  50 

Vol.  II.     Apparatus 2  25 

Vol.  III.     Applications i  50 

Sanford,  P.  G.     Nitro-explosives 8vo,  *4  oo 

Saunders,  C.  H.     Handbook  of  Practical  Mechanics i6mo,  i  oo 

leather,  i   25 

Saunnier,  C.     Watchmaker's  Handbook i2mo,  3  oo 

Sayers,  H.  M.     Brakes  for  Tram  Cars 8vo,  *i  25 

Scheele,  C.  W.     Chemical  Essays 8vo,  *2  oo 

Schellen,  H.     Magneto-electric  and  Dynamo-electric  Machines 8vo,  5  oo 

Scherer,  R.     Casein.     Trans,  by  C.  Salter 8vo,  *3  oo 

Schmall,  C.  N.     First  Course  in  Analytic  Geometry,  Plane  and  Solid. 

i2mo,  half  leather,  *i  75 

Schmall,  C.  N.,  and  Shack,  S.  M.     Elements  of  Plane  Geometry.  ...  i2mo,  *i   25 

Schmeer,  L.     Flow  of  Water 8vo,  *3  oo 

Schumann,  F.     A  Manual  of  Heating  and  Ventilation i2mo,  leather,  i  50 

Schwarz,  E.  H.  L<     Causal  Geology 8vo,  *2  50 

Schweizer,  V.,  Distillation  of  Resins 8vo,  *3  50 

Scott,  W.  W.     Qualitative  Analysis.     A  Laboratory  Manual.  .8vo  (In  Press.) 

Scribner,  J.  M.     Engineers'  and  Mechanics'  Companion  .  . .  i6:no,  leather,  i  50 
Searle,  G.  M.     "  Sumners'  Method."     Condensed  and  Improved.    (Science 

Series.) i6mo,  o  50 

Seaton,  A.  E.     Manual  of  Marine  Engineering 8vo,  6  oo 

Seaton,  A.  E.,  and  Rounthwaite,  H.  M.     Pocket-book  of  Marine  Engineer- 
ing  i6mo,  leather,  3  oo 

Seeligmann,  T.,  Torrilhon,  G.  L.,  and  Falconnet,  H.     India  Rubber  and 

Gutta  Percha.     Trans,  by  J.  G.  Mclntosh 8vo,  *s  oo 

Seidell,  A.     Solubilities  of  Inorganic  and  Organic  Substances 8vo,  *3  oo 

Sellew,  W.  H.     Steel  Rails 4to  (In  Pms.s.) 

Senter,  G.     Outlines  of  Physical  Chemistry i2mo,  *i   50 

Sever,  G.  F.     Electric  Engineering  Experiments 8vo,  boards,  *i  oo 

Sever,  G.  F.,  and  Townsend,  F.     Laboratory  and  Factory  Tescs  in  Electrical 

Engineering 8vo,  *2  50 

Sewall,  C.  H.     Wireless  Telegraphy 8vo,  *2  oo 

-  Lessons  in  Tebgraphv i2mo,  *i  oo 

Sewell,  T.     Elements  of  Electrical  Engineering 8vo,  *3  oo 

-  The  Construction  of  Dynamos 8vo,  *3  oo 

Sexton,  A.  H.     Fuel  and  Refractory  Materials 121110,  *2  50 


D.   VAN   NOSTRAND   COMPANY'S   SHORT  TITLE   CATALOG      19 

Sexton,  A.  H.     Chemistry  of  the  Materials  of  Engineering i2mo,  *2  50 

—  Alloys  (Non- Ferrous) 8vo,  *3  oo 

—  The  Metallurgy  of  Iron  and  Steel 8vo,  *6  50 

Seymour,  A.     Practical  Lithography 8vo,  *2  50 

—  Modern  Printing  Inks 8vo,  *2  oo 

Shaw,  P.  E.     Course  of  Practical  Magnetism  and  Electricity 8vo,  *i  oo 

Shaw,  S.     History  of  the  Staffordshire  Potteries 8vo,  *3  oo 

—  Chemistry  of  Compounds  Used  in  Porcelain  Manufacture 8vo,  *5  oo 

Sheldon,  S.,  and  Hausmann,  E.     Direct  Current  Machines 8vo,  *2  50 

Sheldon,  S.,  Mason,  H.,  and  Hausmann,  E.     Alternating-current  Machines. 

8vo,  *2  50 

Sherer,  R.  Casein.  Trans,  by  C.  Salter 8vo,  *3  oo 

Sherriff,  F.  F.  Oil  Merchants'  Manual i2mo,  *3  50 

Shields,  J.  E.  Notes  on  Engineering  Construction i2mo,  i  50 

Shock,  W.  H.  Steam  Boilers .410,  half  morocco,  15  oo 

Shreve,  S.  H.  Strength  of  Bridges  and  Roofs 8vo,  3  50 

Shunk,  W.  F.  The  Field  Engineer i2mo,  morocco,  2  50 

Simmons,  W.  H.,  and  Appleton,  H.  A.  Handbook  of  Soap  Manufacture. 

8vo,  *3  oo 

Simms,  F.  W.  The  Principles  and  Practice  of  Leveling 8vo,  2  50 

—  Practical  Tunneling 8vo,  7  50 

Simpson,  G.     The  Naval  Constructor i2mo,  morocco,  *5  oo 

Sinclair,  A.     Development  of  the  Locomotive  Engine  .  . .  8vo,  half  leather,  5  oo 

Sindall,  R.  W.     Manufacture  of  Paper.     (Westminster  Series.) 8vo,  *2  oo 

Sloane,  T.  O'C.     Elementary  Electrical  Calculations i2mo,  *2  oo 

Smith,  C.  F.     Practical  Alternating  Currents  and  Testing 8vo,  *2  50 

—  Practical  Testing  of  Dynamos  and  Motors 8vo,  *2  oo 

Smith,  F.  E.     Handbook  of  General  Instruction  for  Mechanics   .  .  .  i2mo,  i  50 
Smith,  I.  W.     The  Theory  of  Deflections  and  of  Latitudes  and  Departures. 

i6mo,  morocco,  3  oo 

Smith,  J.  C.     Manufacture  of  Paint 8vo,  *3  oo 

Smith,  W.     Chemistry  of  Hat  Manufacturing i2mo,  *3  oo 

Snell,  A.  T.     Electric  Motive  Power 8vo,  *4  oo 

Snow,  W.  G.     Pocketbook  of  Steam  Heating  and  Ventilation.    (In  Press.} 
Snow,  W.  G.,  and  Nolan,  T.     Ventilation  of  Buildings.     (Science  Series.) 

i6mo,  o  50 

Soddy,  F.     Radioactivity 8vo,  *3  oo 

Solomon,  M.     Electric  Lamps.     (Westminster  Series.) 8vo,  *2  oo 

Sothern,  J.  W.     The  Marine  Steam  Turbine 8vo,  *5  oo 

Soxhlet,  D.  H.     Dyeing  and  Staining  Marble.     Trans,  by  A.  Morris  and 

H.  Robson 8vo,  *2  50 

Spang,  H.  W.     A  Practical  Treatise  on  Lightning  Protection i2mo,  i  oo 

Speyers,  C.  L.     Text-book  of  Physical  Chemistry SVQ,  *2  25 

Stahl,  A.  W.,  and  Woods,  A.  T.     Elementary  Mechanism i2mo,  *2  oo 

Staley,  C.,  and  Pierson,  G.  S.     The  Separate  System  of  Sewerage.  .  .  .8vo,  *3  oo 

Standage,  H.  C.     Leatherworkers'  Manual 8vo,  *3  50 

—  Sealing  Waxes,  Wafers,  and  Other  Adhesives 8vo,  *2  oo 

—  Agglutinants  of  all  Kinds  for  all  Purposes i2mo,  *3  50 

Stansbie,  J.  H.     Iron  and  Steel.     (Westminster  Series.) 8vo,  *2  oo 

Stevens,  H.  P.     Paper  Mill  Chemist i6mo,  *2  50 

Stewart,  A.     Modern  Polyphase  Machinery i2mo,  *2  oo 


20      D.   VAN   NOSTRAND   COMPANY'S   SHORT  TITLE  CATALOG 

Stewart,  G.     Modern  Steam  Traps i2iro,  *i  25 

Stiles,  A.     Tables  for  Field  Engineers 121110,  i  oo 

Stillman,  P.     Steam-engine  Indicator - i2mo,  i  oo 

Stodola,  A.     Steam  Turbines.     Trans,  by  L.  C.  Loewenstein 8vo,  *5  oo 

Stone,  H.     The  Timbers  of  Commerce , 8vo,  3  50 

Stone,  Gen.  R.    New  Roads  and  Road  Laws. i2mo,  i  oo 

Stopes,  M.     Ancient  Plants 8vo,  *2  oo 

Sudborough,  J.  J.,  and  James,  T.  C.     Practical  Organic  Chemistry.  .  i2mo,  *2  oo 

Suffling,  E.  R.     Treatise  on  the  Art  of  Glass  Painting 8vo,  *3  50 

Swan,  K.     Patents,  Designs  and  Trade  Marks.     (Westminster  Series.) .  8vo,  *2  oo 

Sweet,  S.  H.     Special  Report  on  Coal 8vo,  3  oo 

Swoope,  C.  W.     Practical  Lessons  in  Electricity i2mo,  *2  oo 

Tailfer,  L.     Bleaching  Linen  and  Cotton  Yarn  and  Fabrics 8vo,  *5  oo 

Templeton,  W.     Practical  Mechanic's  Workshop  Companion. 

i2mo,  morocco,  2  oo 
Terry,  H.  L.     India  Rubber  and  its  Manufacture.     (Westminster  Series.) 

8vo,  *2  oo 

Thorn,  C.,  and  Jones,  W.  H.     Telegraphic  Connections oblong  i2mo,  i  50 

Thomas,  C.  W.     Paper-makers'  Handbook (In  Press.) 

Thompson,  A.  B.     Oil  Fields  of  Russia 4to,  *7  50 

—  Petroleum  Mining  and  Oil  Field  Development 8vo,  *5  oo 

Thompson,  E.  P.     How  to  Make  Inventions 8vo,  o  50 

Thompson,  W.  P.     Handbook  of  Patent  Law  of  All  Countries i6mo,  i  50 

Thornley,  T.     Cotton  Combing  Machines 8vo,  *3  oo 

—  Cotton  Spinning.     8vo. 

First  Year *i  50 

Second  Year *2  50 

Third  Year *2  50 

Thurso,  J.  W.     Modern  Turbine  Practice 8vo,  *4  oo 

Tinney,  W.  H.     Gold-mining  Machinery 8vo,  *5  oo 

Titherley,  A.  W.     Laboratory  Course  of  Organic  Chemistry 8vo,  *2  -oo 

Toch,  M.     Chemistry  and  Technology  of  Mixed  Paints 8vo,  *3  oo 

Todd,  J.,  and  Whall,  W.  B.     Practical  Seamanship 8vo,  *y  50 

Tonge,  J.     Coal.     (Westminster  Series.) 8vo,  *2  oo 

Townsend,  J.     lonization  of  Gases  by  Collision 8vo,  *i  75 

Transactions  of  the  American  Institute  of  Chemical  Engineers.     8vo. 

Vol.    I.     1908 *6  oo 

Vol.  II.     1909 *6  oo 

Traverse  Tables.     (Science  Series.) i6mo,  o  50 

morocco,  i  oo 

Trinks,  W.,  and  Housum,  C.     Shaft  Governors.     (Science  Series.) .  .  i6mo,  o  50 

Tucker,  J.  H.     A  Manual  of  Sugar  Analysis 8vo,  3  50 

Tumlirz,  O.     Potential.     Trans,  by  D.  Robertson i2mo,  i  25 

Tunner,  P.  A.     Treatise  on  Roll-turning.     Trans,  by  J.  B.  Pearse. 

8vo,  text  and  folio  atlas,  10  oo 

Turbayne,  A.  A.     Alphabets  and  Numerals 4to,  2  oo 

Turrill,  S.  M.     Elementary  Course  in  Perspective i2mo,  *i  25 

Underbill,  C.  R.     Solenoids,  Electromagnets  and  Electromagnetic  Wind- 
ings  i2mo,  *2  oo 


D.   VAN    NOSTRAND   COMPANY'S  SHORT  TITLE   CATALOG      21 

Urquhart,  J.  W.     Electric  Light  Fitting i2mo,  2  oo 

—  Electro-plating • i2mo,  2  oo 

—  Electrotyping i2mo,  2  oo 

—  Electric  Ship  Lighting • i2mo,  3  oo 

Universal  Telegraph  Cipher  Code i2mo,  i  oo 

Vacher,  F.     Food  Inspector's  Handbook i2mo,  *2  50 

Van  Nostrand's  Chemical  Annual.     Second  issue  1909 i2mo,  *2  50 

-  Year  Book  of  Mechanical  Engineering  Data.     First  issue  1910.  .  .  (In  Press.) 

Van  Wagenen,  T.  F.     Manual  of  Hydraulic  Mining i6mo,  i  oo 

Vega,  Baron  Von.     Logarithmic  Tables 8vo,  half  morocco,  2  50 

Villon,  A.  M.     Practical  Treatise  on  the  Leather  Industry.     Trans,  by  F. 

T.  Addyman 8vo,  *io  oo 

Vincent,  C.  Ammonia  and  its  Compounds.  Trans,  by  M.  J.  Salter .  .  8vo,  *2  oo 

Volk,  C.  Haulage  and  Winding  Appliances 8vo,  *4  oo 

Von  Georgievics,  G.  Chemical  Technology  of  Textile  Fibres.  Trans,  by 

C.  Salter 8vo,  *4  50 

—  Chemistry  of  Dyestuffs.     Trans,  by  C.  Salter 8vo,  *4  50 

Wabner,  R.     Ventilation  in  Mines.     Trans,  by  C.  Salter 8vo,  *4  50 

Wade,  E.  J.     Secondary  Batteries 8vo,  *4  oo 

Wadsworth,  C.     Primary  Battery  Ignition i2mo  (In  Press.) 

Wagner,  E.     Preserving  Fruits,  Vegetables,  and  Meat i2mo,  *2  50 

Walker,  F.,    Aerial  Navigation 8vo,  3  oo 

—  Electric  Lighting  for  Marine  Engineers 8vo,  2  oo 

Walker,  S.  F.     Steam  Boilers,  Engines  and  Turbines 8vo,  3  oo 

—  Refrigeration,  Heating  and  Ventilation  on  Shipboard. 

i2mo  (In  Press.) 

—  Electricity  in  Mining 8vo,  *3  50 

—  Steam  Boilers,  Engines  and  Turbines 8vo,  *3  oo 

Walker,  W.  H.     Screw  Propulsion 8vo,  o  75 

Wallis-Tayler,  A.  J.     Bearings  and  Lubrication 8vo,  *i  50 

—  Modern  Cycles 8vo,  4  oo 

—  Motor  Cars 8vo,  i  80 

—  Motor  Vehicles  for  Business  Purposes '. 8vo,  3  50 

—  Pocket  Book  of  Refrigeration  and  Ice  Making i2mo,  i  50 

—  Refrigerating  and  Ice-making  Machinery .  8vo,  3  oo 

—  Refrigeration  and  Cold  Storage 8vo,  *4  50 

—  Sugar  Machinery i2mo,  *2  oo 

Wanklyn,  J.  A.     Treatise  on  the  Examination  of  Milk i2mo,  i  oo 

-  Water  Analysis i2ino,  2  oo 

Wansbrough,  W.  D.     The  A  B  C  of  the  Differential  Calculus i2mo,  *i  50 

—  Slide  Valves i2mo,  *2  oo 

Ward,  J.  H.     Steam  for  the  Million 8vo,  i  oo 

Waring,  G.  E.,  Jr.     Sewerage  and  Land  Drainage *6  oo 

--  Modern  Methods  of  Sewage  Disposal i2mo,  2  oo 

—  How  to  Drain  a  House i2mo,  i  25 

Warren,  F.  D.     Handbook  on  Reinforced  Concrete i2mo,  *2  50 

Watkins,  A.     Photography.     (Westminster  Series.) 8vo  (In  Press.) 

Watson,  E.  P.     Small  Engines  and  Boilers i2mo,  i  25 

Watt,  A.     Electro-plating  and  Electro-refining  of  Metals 8vo,  *4  50 


22      D.   VAN   NOSTRAND   COMPANY'S   SHORT  TITLE   CATALOG 

Watt,  A.     Electro-metallurgy i2mo,  i  oo 

The  Art  of  Soap-making • 8vo,  3  oo 

—  Leather  Manufacture 8vo,  *4  oo 

Weale,  J.     Dictionary  of  Terms  Used  in  Architecture i2mo,  2  50 

Weather  and  Weather  Instruments i2ino,  i  oo 

paper,  o  50 

Webb,  H.  L.  Guide  to  the  Testing  of  Insulated  Wires  and  Cables. .  i2mo,  i  oo 

Webber,  W.  H.  Y.  Town  Gas.  (Westminster  Series.) 8vo,  *2  oo 

Weekes,  R.  W.  The  Design  of  Alternate  Current  Transformers. .  ..i2mo,  i  oo 

Weisbach,  J.  A  Manual  of  Theoretical  Mechanics 8vo,  *6  oo 

sheep,  *7  50 

Weisbach,  J.,  and  Herrmann,  G.  Mechanics  of  Air  Machinery 8vo,  *3  75 

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